Diffusion Indexes:

The Case of Singapore

Lim You Jie, Benedict

May 30, 2010

Abstract

This paper uses factor analysis, a technique that allows researchers to utilize large

cross-sectional macroeconomic data, for the forecasting of inflation and the

production index in Singapore. This paper follows closely the techniques used in

Stock and Watson’s paper on Diffusion Indexes (1998), and utilizes updated

factor selection techniques proposed by Bai and Ng (2001). The factor models

used in this study contains 155 quarterly series observed over the period 1992:Q2

to 2009:Q4. The results based on the RMSE of the 1, 2, 3, 4, and 8 period ahead

forecasts indicate that factor models outperform other conventional forecasting

models such as the random walk model, autoregressive model as well as the

Phillips Curve at the 4 and 8 quarter ahead horizon.

Keywords: Factor Model, AO Random Walk, Diffusion Indexes, Singapore, Forecast Accuracy.

_____________________________________

I am grateful to my thesis advisor Professor Giorgio Primiceri for patiently advising me during the

process of writing this thesis. I am also thankful to Hyerim Shin for her assistance with MATLAB coding.

1. Introduction

Diffusion indexes are averages of the contemporaneous values of a large number of time

series such as the gross domestic product, unemployment rate and money supply. These indexes

have been historically calculated by economists who use their expert opinion to decide what time

series should be included in the indexes and how much weight should be allocated to each series.

These indexes are then used to allow policymakers to identify turning points in the economy

such as economic upswings or recessions. A drawback to this method of constructing diffusion

indexes is that the economist’s beliefs are projected into his or her selection criterion. In this

paper, we use a factor analysis, a technique that does not impose a priori assumptions and

economic theory, to construct diffusion indexes for the Singapore economy.

Accurate forecasts play an integral role in how businesses plan their future expenditures and

financial strategies, how individuals adjust their savings, and also how investors adjust their

expectations for stock markets. For policy makers, central banks need accurate inflation forecasts

to allow them to analyze the potential ramifications of their monetary policies, and also to allow

them to analyze what inflation would look like without any governmental intervention. Accurate

forecasts of the Industrial Production Index (IPI) are also important because it is the most

reliable quantitative indicator on economic activity, allowing economists to better analyze

business cycles.

Figure 1.

Figure 2.

From Figures 1 and 2 graphs the quarterly annualized inflation and IPI growth

respectively. We see that Singapore generally has had low and stable inflation, while IPI growth

has been very volatile, ranging from 40% to negative 48% growth.

The ability to achieve accurate forecasts in Singapore is even more pressing because of

its status as a country with a small open economy. As an island state without natural resources,

Singapore is heavily dependent on trade, as seen by the fact that the sum of merchandise imports

and exports is 362% as a ratio of GDP (World Bank, World Development Indicators). Thus it is

not surprising that the Monetary Authority of Singapore (MAS) follows an exchange-rate centric

monetary policy, and has helped Singapore maintain low inflation rates and sustained economic

growth in the last 45 years of independence. The exchange rate is pegged to an undisclosed

trade-weighted basket of currencies, and adjustments to the exchange rate are dependent mainly

on expected inflationary pressures. Any misjudgments and policy misstep would have serious

repercussions on Singapore’s economy.

The main goal of this paper is to assess if the use of factor models for forecasting actually

improves upon the accuracy of inflation and IPI growth forecasts when compared to other

conventional forecasting models. The other models that are used for comparison are the Atkeson-

Ohanian (2001) random walk model, direct autoregressive model, multivariate leading indicator

model, and the Phillips Curve model. We find that factor models do indeed show a significant

improvement in only inflation forecasting accuracy as compared to the competing models at the

4 and 8 quarter ahead forecasting horizon. Factor models do not improve the forecasting

performance of IPI growth forecasts. Moreover, when using the Diebold-Mariano (1995) and

Giancomini-White (2003) test to test for the statistical significance of this improvement in

forecasting accuracy, we find that this improvement is generally significant only at the 8 quarter

ahead horizon. And lastly, we do not find any conclusive improvement in forecast accuracy

when implementing the Bai-Ng (2002) information criterion in the selection of the optimal

number of factors to be included in the forecasting model.

The paper is structured as such: The second section of the paper surveys the related

existing literature. The third section gives a detailed list of the macroeconomic variables used in

the paper and its associated transformations. Section four presents the theoretical factor model

and the algorithms used to derive the common factors, as well as the methodology used to

determine the optimal number of common factors. Section five describes the other forecasting

models. The sixth section gives a brief description of the general forecast methodology. In

section seven we detail and analyze the empirical results from all of our forecasting models, and

we lastly conclude in section eight.

2. Literature Review

Factor models have been used in research since Sargent and Sims (1977) as well as

Geweke (1977) used factor analysis as a way to analyze macroeconomic activity without

imposing assumptions and economic theory a priori. Their main contribution was to discover that

with the use of factor models, a small number of factors could be extracted to account for the

majority of the variation observed in major economic aggregates. Recently, there has been an

increase in the use of factor models in forecasting around the world, spearheaded by James H.

Stock and Mark W. Watson in a series of papers (1998, 1999, 2002) that analyzed the viability of

utilizing factor models for forecasting inflation and IPI in the United States. The recent

renaissance in factor analysis as an approach towards macroeconomic forecasting can be

attributed to breakthroughs such as the advancement of estimation techniques developed by

Stock and Watson (1998), where they used the principal components method to estimate factor

model parameters. Moreover, the availability of a wide variety of macroeconomic data due to

increased use of computers, recent computational advancements that allowed researchers to

process large volumes of data, and the appeal of driving forecasts based on a large number of

variables without any a priori assumptions, all made factor models more accessible to both

researchers and policymakers.

Many economists and policymakers around the world have conducted empirical studies

utilizing factor models for forecasting, for example Gosselin and Tkacz Kotlowski (2008) in the

case of Poland, Gosselin (2001) in the case of Canada, Kunovac (2007) in the case of Croatia,

and also Angelini, Henry and Mestre (2001) in the case of the Euro area. There have not been

many papers written to analyze how factor models can be used in the context of Singapore, but

the most notable would be a paper by Chow and Choy (2009) where the authors utilized a

dynamic factor model to analyze and forecast business cycles in Singapore. This paper differs

from Chow and Choy (2009) in that we used a modified and updated dataset, we took a different

approach in the factor model forecast methodology, and included a larger variety of inflation

forecasting models for comparison so as to better evaluate the effectiveness of factor models in

forecasting.

The resurgence of factor models as a forecasting tool also drove research on how

effective factor models actually are. Stock and Watson (1999a, 2002, 2004) as well as Forni et el

(2001, 2005) compared factor models to other forecast methods and concluded that factor models

did indeed outperform various benchmark forecasting models, especially against small-scale and

simpler forecasting models. However, Groen, Kapetanios and Price (2007), Faust and Wright

(2007), Eickmeier and Ziegler (2006), Schumacher and Dreger (2006) and Gosselin (2001) also

argued that factor models did not necessarily improve forecast performances when compared to

other existing complex large scale data forecasting models, that the major improvement in

forecasting ability of factor models was statistically insignificant. However, the general outlook

on the usefulness of factor models is generally positive, and current research is focused on

improving factor model forecasts in areas such as non-parametric analysis as well as the in the

area of how to determine optimal number of factors.

In this paper, it was imperative to select an appropriate model as a benchmark model

against which factor forecasting models for inflation and IPI could be compared against. The

need for an alternative reliable inflation forecasting model arose when Jaditz and Sayers (1994)

first noted the weakening of inflation forecast capabilities of the Phillips Curve for the United

States since the mid-1980s. Many other similar empirical studies (Stock and Watson 1998, 2008,

Orphanides 2004, Atkeson-Ohanian 2001) also concluded that while the Phillips Curve model

had been a reliable model for policymakers prior the mid-1980s, other forecasting models

performed better than the traditional Phillips Curve during this period. Atkeson-Ohanian (2001)

in particular showed in an extreme case that modern Phillips Curve based models did not

outperform a simple random walk model, a model akin to what they described as predicting

inflation based on a coin-flip.

Further inflation forecasting empirical studies in the case of the United States showed that the

random walk model not only outperformed the Phillips Curve models, but also other more

sophisticated models. In light of the Atkeson-Ohanian (2001) random walk model's strong

performance in inflation forecasting in the United States for the last 20 years, and because the

model does not place any prior assumptions or theory on inflation forecasting, the random-walk

model is an important benchmark model, allowing us to perform an impartial evaluation.

Conceptually, there are no reasons the Atkeson-Ohanian (2001) random walk model cannot be

extended to be used for IPI forecasting. Thus in this paper, we extend use the Atkeson-Ohanian

(2001) random walk model as the main benchmark to compare forecast performance for both

inflation and of the Industrial Production Index.

3. Description of the Data

Since Singapore's independence in 1965, Singapore has maintained an updated,

extensive, and reliable statistics database on the economy and population. The data used in this

study consist mainly of macroeconomic variables specific to the Singapore economy, as well as

data on major macroeconomic variables on foreign countries with strong economic ties to

Singapore. The selection of time-series was closely based on the 132 variables used in the study

by Stock and Watson (2002) for the US economy, as well as the variables Chow and Choy

(2009) utilized in their dynamic factor modeling exercise for the Singapore economy. Departing

from the two papers, we placed an additional emphasis on trade related variables because of the

economy’s heavy dependence on trade as a global trading hub.

Out of the numerous variables initially under consideration, we dropped variables based

on their frequency and the earliest availability of observations so that there would be no

unobserved variables. Monthly data were either aggregated or averaged to obtain data in the

quarterly format depending on the type of time series. The final list of variables selected

consisted of monthly and quarterly data encompassing the period from 1992:2nd

Quarter to

2009:4th

Quarter (71 Observations). In all, 155 variables were used in the study, and the balanced

dataset was then categorized into several macroeconomic groups to facilitate analysis. The 15

macroeconomic groups are: Singapore real GDP and expenditure components, trade indicators,

general price indices, sectorial indicators, labor market variables, construction sector, industrial

production indices, monetary indicators, financial indicators, business expectations, foreign stock

exchange prices, foreign composite leading index, and lastly foreign GDP.

The time series were then transformed with a similar approach that Stock and Watson

(2005) used so that the time series were approximately stationary and thus could be appropriately

used in factor model specification. Generally, the first difference of logarithms was taken for real

quantity variables to get the quarterly percentage change, second difference of logarithms was

taken for price series and monetary indicators, and the first difference was used for nominal

interest rates. Series that were in the form of percentage changes were not adjusted. After the

dataset was transformed, the dataset was adjusted for outliers following Stock and Watson

(2005) by first identifying and then replacing observations with absolute median deviations 6

times larger than the 75th

and 25th

inter quartile range with the median of the previous 5

observations. After de-trending and adjusting the dataset for outliers, the series were then

normalized to have zero mean and unit standard deviation (Stock and Watson 1998). A

comprehensive list of the variables utilized in this paper and applied transformations are listed in

greater detail in Appendix A.

4. The Factor Model

4.1 The Exact Static Factor Model

In this paper, we follow the procedure used by Stock and Watson (1998 and 2002), where

the forecasting is performed using a two-step process. The first step involves estimating the

common factors from the dataset, and the second step involves performing a linear regression of

the inflation rate on the factors. We use the factor model proposed by Stock and Watson (2002)

to estimate the common factors driving most of the variability observed in the data. We assume

that , the vector time series variable that contain useful information for forecasting , can

be represented by:

(1)

where is a (r x 1) common factor, r is the number of common static factors, is a (N x 1)

idiosyncratic disturbance vector, and Λt is the (N x r) factor loading matrix.

To see how the common factors are implemented in this forecasting exercise, we briefly

describe the general forecast model as:

(2)

where is the scalar time series variable to be forecasted, h is the forecast horizon, is the

common factors extracted from (1), is a (m x 1) vector of observed variables (which in this

paper are the lags of ), and is the forecast error.

In this paper, we make the assumptions that eit and ejt for i ≠ j, as well as are mutually

uncorrelated and are i.i.d. We also assume that the factor loadings are constant, where .

All these assumptions combined make the model an exact static factor model and allow us to

estimate the common factors and factor loadings.

4.2 Other Variations of the Factor Model

There are also variations of the factor model that differs in the underlying assumptions

made. One variation is where the idiosyncratic disturbances are allowed to be serially correlated

and cross-sectionally correlated. When these errors are allowed to be weakly correlated, we call

this model an approximate static factor model (Chamberlain and Rothschild 1984, Stock and

Watson 1998).

Another variation is the dynamic factor model which we do not use in this paper is where

we allow lags of the factors to enter the equation, thus we first write the model as:

(3)

(4)

where and are lag polynomials with finite order q, which is the number of dynamic

factors. With appropriate manipulation of equations (3) and (4), Stock and Watson (2002)

showed that the dynamic factor model can be rewritten in the static form of (1) and (2).

(5)

Where ( ) and

4.2 Estimation of the Factors Using Principle Components

Different methods have been proposed to extract the common factors in factor models.

One method proposed by Forni, Hallin, Lippi and Reichlin (2000, 2003), is a two step approach

that finds the solution of a generalized principle component problem. First the factor estimates

are derived from the spectral density matrices of the time series, these estimates are then used to

construct contemporaneous linear combinations of observed data so as to minimize the

idiosyncratic-common variance ratio. In this paper, we use the method of principal components

proposed by Stock and Watson (1998) to extract the common factors.

The static representation of the factor model in (1) allows us, by the method of principal

components, to non-parametrically estimate the factors. We need to solve for the minimum of the

nonlinear least squares objective function of:

( ) ∑ ∑

(6)

Let ( denote the minimizers of (6), and these estimates satisfy the first order

conditions:

∑

∑

(7)

∑

∑

(8)

Solving for the minimum of (6) by substituting (7) or (8) will yield two different

eigenvalue problems, and each problem can be solved by choosing the factors as the

eigenvectors corresponding to the k largest eigenvalues of the matrix ∑

, or the matrix

∑

respectively, where k is the number of factors desired. Thus the algorithm to find the

common factors can be summarized as such: Given a balanced normalized dataset, we find the

matrix of eigenvectors and the corresponding diagonal matrix of eigenvalues where the

eigenvalues have been arranged in decreasing order, then selected the required number of

common factors from the matrix of eigenvectors. This procedure was programmed in MATLAB

to allow for a recursive out of sample forecasting exercise.

4.3 The Determination of Number of Factors using the Bai-Ng Test

We take two approaches to determine the number of factors to be used in our forecasting

models. The first approach is to test the performance of forecasting models with a fixed number

of variables, so as not to impose any prior methodology of determining the optimal factors to

include in the forecasting model. The second approach is the implementation of a selection

criterion to determine the optimal number of lags we want to include in our forecasting models.

There have been not many tests proposed to determine the number of common factors

driving the factor model. One popular criterion is the Bai-Ng (2002) test, which Stock and

Watson (2010) conducted limited Monte Carlo experiments upon and concluded that it had better

finite sample performance when compared with another popular alternative, the Amenguel-

Watson (2007) procedure. In this paper, the method we utilize is the Bai-Ng (2002) test to isolate

the number of common factors in the dataset.

In Stock and Watson (1998), the author used the Bayesian Information Criterion (BIC) to

determine the number of factors to include in their forecasting model. But Bai & Ng (2002)

showed that the BIC and Akaike’s Information Criterion (AIC) would overestimate the number

of factors required for factor analysis because the penalty function for over fitting was only a

function the number of observed periods T, and not of both T and the number of variables N.

Bai & Ng (2002) wanted an information criterion (IC) that consisted of two parts like (9).

The first term is such that the average residual variance decreases with increasing number of

factors, while the second term is a penalty function imposed on the IC that increases

with the number of factors due to over identification.

( ) (9)

Bai & Ng (2002) then proposed three different Information Criterions (IC) to determine

the optimal number of static factors:

( ( )) (

) (

)

(10)

( ( )) (

)

(11)

( ( ))

(12)

where {√ √ }, and ( )

∑ ∑

from equation (6)

Even though the three criteria are asymptotically equivalent, they have different

properties in finite samples; in fact, only (10) and (11) apply specifically to principal

components. Because Bai & Ng (2002, 2007) found that was effective as a test itself, we

utilized only when utilizing the Bai & Ng test in determining the number of factors. We

estimate the number of static factors in the model by choosing k such that , which

Bai & Ng (2002) demonstrated consistently estimates r, the true number of static factors.

5. Alternative Forecasting Models

The objective of this paper is to test the accuracy of factor model forecasts for the annual

inflation and IPI growth rate 1, 2, 3, 4 and 8 periods ahead. For notational purposes, we denote

the h-step ahead annualized inflation rate (where h = 1, 2, 3, 4, and 8) as:

[ (

)]

(13)

[ (

)]

.

(14)

where is the CPI at time t in the case of inflation forecasting, and is the Industrial

Production Index at time t in the case of forecasting IPI growth rates. is the h-step ahead

forecast of the target growth rate.

5.1 Atkeson-Ohanian (AO) Random Walk Model (Both Inflation and IPI Forecasting)

The AO random walk model is a univariate model that predicts that the forecast of the annual

rate of inflation or IPI growth rate is simply the average growth rate of the target variable over

the previous four quarters. Though Atkeson and Ohanian (2001) only considered the 4 period

ahead forecasts their random walk model, we extend this model to forecast 1, 2, 3, 4 and 8

quarters ahead. For example, if the average inflation for the end of the four quarters for the year

of 2002 was 3%, the forecasts for the next 1, 2, 3, 4 and 8 quarters ahead forecast would be 3%.

5.2 Direct Autoregressive (AR) Model (Both Inflation and IPI Forecasting)

The direct autoregressive forecast is obtained by ordinary least squares from the model:

∑

(15)

where h denotes the number of periods ahead we are forecasting, is a constant, denotes

the quarterly inflation rate, and is the h step ahead error term. Two methods were used to

determine the number of lags in the AR model: In the first model, the lag length at every forecast

period in the AR model was determined by the Bayesian Information Criterion with a maximum

of 6 lags. In a second set of models, a fixed number of one to six lags were used to calculate the

forecasts.

5.3 Multivariate Leading Indicator Model (Only Inflation Forecasting)

The multivariate leading indicator model is stated as:

∑

∑

(16)

where the parameters are estimated using ordinary least squares. h denotes the number of periods

ahead we are forecasting, denotes the quarterly inflation rate, }is a vector of the leading

indicators, is the h step ahead error term, and number of lags q and p were selected

separately by BIC (maximum lag of 6). The leading indicators were chosen following Stock and

Watson (1999) where the authors chose indicators that were proven to be effective for

forecasting purposes. We then selected leading indicators for Singapore based on Stock and

Watson (1999) as closely as possible where data was available. The leading indicators used are

the unemployment rate, forecasts of total new orders received, value of construction contracts

awarded, M1 money supply, overnight interbank rate, spread on overnight rate and T-bill, and

lastly the US dollar exchange rate.

5.4 Phillips Curve Model (Only Inflation Forecasting)

The Phillips Curve was first proposed by the New Zealand economist William Phillips, a

model that proposes an inverse and stable relationship between the level of unemployment and

the inflation level. The Phillips Curve gained popularity due to the policy implications the model

presented: If the Phillips Curve were true, then the government could target a certain level of

unemployment as means of ensuring price stability. Samuelson and Solow (1960) presented the

idea of a trade-off between unemployment and inflation, a trade-off policy makers could

capitalize on to achieve their macroeconomic goals. The original Phillips Curve model has been

modified over time to better explain observations of inflation and to improve inflation forecasts.

Criticism of the inability of the Phillips Curve to make reliable forecasts spurred economists

to make constant refinements to their models to try to account for policy and structural changes.

The Expectations-Augmented Phillips Curve and the Triangle model proposed by Gordon (1991)

are examples of such modifications of the original Phillips Curve. We use the second prototype

Phillips Curve model utilized in Stock and Watson (2008) which is an Autoregressive

Distributed Lag (ADL) model. The model is represented as:

∑

∑

(17)

where the parameters are estimated using ordinary least squares. h denotes the number of periods

ahead we are forecasting, denotes the quarterly inflation rate, the unemployment rate,

is the h step ahead error term, and number of lags q and p were selected separately by BIC

(maximum lag of 6).

5.5 Factor Model (Both Inflation and IPI Forecasting)

This model follows closely the diffusion index forecasting model used in Stock and Watson

(1998) of the form:

∑

∑

(18)

where { } are the estimated factors, is the quarterly growth rate at the period t-j, and is

the error term. The estimates for the coefficients of the diffusion index forecasting model were

calculated using OLS. We depart from Stock and Watson (1998) in using the Bai-Ng (2002) test

to select the optimal number of factors in the factor model instead of using BIC. We consider

four versions of (2):

i) The number of factors q in the forecasting model chosen at each period by the Bai-Ng

test, where , and no autoregressive components (p=0)

ii) The number of factors q recursively chosen by the Bai-Ng test, where ,

and the number of autoregressive components recursively chosen by BIC ( )

iii) No autoregressive components (p=0) but a fixed number of factors ( )

iv) A fixed number of factors( ) and autoregressive component p selected by

BIC ( )

6. Forecast Methodology

6.1.1 Forecast Comparison using the Giacomini-White (2006) Test

In order to evaluate whether the observed differences between the RMSEs calculated

from the forecasts are statistically significant, we implement a forecast evaluation to assess if a

particular forecast model statistically outperforms a benchmark model. Numerous tests have

been proposed by Diebold and Mariano (1995), West (1996), Clark and McCraken (2004), and

Giacomini and White (2006) to compare out of sample predictive ability for different forecasting

conditions. Popular tests such as the Diebold and Mariano (1995) and the West (1996) tests are

not directly applicable because these tests are applicable only in the comparison of non-nested

models. To best evaluate the statistical significance of the forecast performances of two

competing models, use the Giacomini-White (2006) test because it is a general test that permits a

comparison of both nested and non-nested models.

The test checks if there is a statistically significant difference in the forecasting

performance between two models. Thus the null hypothesis for the Giacomini-White equal

conditional predictive ability test is written as:

[ ( ) ( )| ] [ ] (19)

where is a loss function which in this case is the squared error, is the forecast horizon,

and are the two forecast model functions.

against the alternative:

for all sufficiently large n (20)

where ∑ , and is the test function. The alternative hypothesis can be

generally interpreted as the case where the differences in forecasting performance are not

statistically significant. In this paper, we use the same test function, , that

Giancomini & White (2006s) used in their tests to determine if the predicted diffusion index

forecasts that Stock and Watson (2002) generated were statistically significant. The flexibility1 in

selecting the test function allows the researcher to input his beliefs on how past relative

performance may help distinguish between the forecast performance in the future. In using

, we impose the beliefs that the difference in predictive ability today has potential

explanatory power for the future difference.

Lastly, the test statistic to be constructed is:

∑

∑

(21)

where n is the number of observations in the forecast period, ∑

∑ ∑

with a weight function such as

the one in Newey and West (1987). is also a consistent estimator of the variance of .

Lastly, the test statistic follows a distribution, which allows us to calculate the p-values of the

tests for statistical difference in forecasting performances.

6.1.2 Forecast Comparison using the Diebold-Mariano (1995) Test

Though the Diebold-Mariano (1995) test only allows us to make comparisons between non-

nested models, many empirical studies still implement the Diebold-Mariano (1995) test. It is

arguable that the AO Random-Walk model when compared to the other competing models is not

nested, thus the results of statistical significance obtained from this test should be accepted with

caution. Furthermore, since our forecast period is relatively short, we use Harvey, Leybourne and

Newbold (1997)’s small sample modification of the Diebold-Mariano (1995) test to evaluate

statistical significance.

1 In a paper where factor models were used to forecast inflation in Mexico (Ibarra-Ramirez 2010),

the author used a constant as an instrument when implementing the Giancomini & White (2003)

test.

The null hypothesis for the Diebold-Mariano (1995) test can be described as the equality

between the squared errors obtained from the two forecasting models being compared, and can

be stated as:

[ ] (22)

where is the same loss difference in the previous section.

We first define

∑

(23)

[ ∑

] (24)

∑

The modified Diebold-Mariano (1995) test statistic can then be written as:

(25)

[

]

[ ]

[

]

(26)

where n is the number of observations, is the kth

autocovariance of , and is the number

of periods ahead which we are forecasting. This test statistic can be compared against the critical

values from the Student’s t-distribution with degrees of freedom to allow us to calculate

the p-values of the tests for statistical difference in forecasting performances.

6.2 Estimation

In order to obtain the estimates of the forecasts from the model, we use the ‘direct’ forecast

method in which we regress a multiperiod-ahead value of the dependent variable on available

data from the past till today. Optimally, we would want to perform real-time forecasts, but such

forecasts are prohibitively complicated because of the large dataset involved which are often

announced with different lags and are subject to revision subsequently. However, we attempt to

closely replicate a real-time forecasting exercise by performing out-of-sample direct forecasts.

In the case of factor model forecasting, in order to obtain the forecast of period T+h where h

is the number of periods ahead we want to forecast, the data from period 1 to T was normalized

to avoid overweighing any variable in the dataset. After normalizing the data, the factors were

estimated by the method of principle components, the optimal number of factors and lags

selected by the Bai-Ng test and BIC respectively, and the regression coefficients were then

estimated from time 1 to T using OLS. The forecast was then constructed using the estimated

parameters and factors, and finally compared to the actual values for inflation at the forecast

horizon using an appropriate metric. To forecast the next period T+1+h, we repeat the procedure

using data from period 1 to T+1, reevaluating the procedure of calculating the information

criterions, factors, and estimators before making the forecast. The Diebold-Mariano (1995) test is

used in the fixed window estimation to evaluate the statistical significances of forecasting

performances.

The implementation of the Giacomini-White (2006) test described in section 6.1 required

the forecasting model to depend on forecasts generated on rolling window scheme. Instead of

adding a new observation to our dataset as in our first estimation exercise, we fix the estimation

window to 32 quarters in the rolling window exercise. For example, if we utilized data from

period 1 to period T to estimate forecasts at time T+h, we move the estimation window 1 quarter

ahead to data from time 2 to time T+1 to estimate forecasts at time T+h+1. The procedure to

calculating the information criterions, factors and estimators is similar to the previous exercise,

only the data used in the forecasts varies.

7. Results

7.1 Analyzing the Common Factors Extracted

Table 1 shows the values of the eigenvector and variance associated with factors 1

through 12. On the third column it also describes the cumulative percentage of variance

explained. From table 1 we can see that with the first 6 factors extracted from the

macroeconomic variables, we are able to explain almost half of the variation of our data set.

Moreover, the amount of variance explained by each factor is heavily skewed towards the first

three factors, explaining a cumulative 33.6% of total variance observed2. This result suggests that

there are only a few variables that are the source of most of the macroeconomic variability found

in Singapore, which leads us to try to identify the main sources of these variations.

Table 1. Variance explained by the first r common factors

r Eigenvector

value

Variance explained

by factor (%)

Cumulative percentage of

variance explained (%)

1 28.541 17.727 17.727

2 15.036 9.339 27.066

3 10.522 6.536 33.602

4 8.066 5.010 38.612

5 7.353 4.567 43.179

6 6.677 4.147 47.327

7 5.455 3.388 50.715

2 Similarly, a formal method utilizing a scree plot plots the eigenvectors for each factor in descending order, and the

optimal number of factors to be chosen is determined by the factor at the kink.

8 5.104 3.170 53.885

9 4.707 2.924 56.808

10 4.216 2.618 59.427

11 3.919 2.434 61.860

12 3.568 2.216 64.077

The factors extracted from the factor model itself do not have explicit interpretations, and

moreover could just be linear combinations of the variables from the time series. But because

these factors span the same space as the structural factors (Kotlowski 2008), we adopt Stock and

Watson’s (2002b) methodology of regressing each of the individual time series against each of

the extracted common factors and finding the coefficient of determination, or , for each

regression. Stock and Watson (2002b) used the values of the associated with each variable to

evaluate the composition of each factor; the higher the values of , the better the particular

factor explains a specific variable. In Figure 3, the s of the individual variables corresponding

to the top 5 common factors are plotted. On the horizontal axis are the 161 variables, and on the

vertical axis is the value of the R2s. The vertical lines that partition the bars separate the variables

separate into the general sectors that were previously defined in section 3.

Figure 3. From the Regressions of the Variables on the Top 5 Factors

From figure 3, we see that Factor 1 is characterized mainly by three groups of

macroeconomic time series: The bulk of the factor is driven by trade related variables such as

import and export of goods and services, as well as economic indicators such as the Composite

Leading Index and GDP of Singapore’s major trading partners. Lastly, the factor is also driven

by business expectations of the Singapore economy, for example the variables on expectations

on the manufacturing sector and the total new orders received. The characterization of the main

factor driving the variability reflects Singapore’s economy that is historically and currently

driven by exports and manufacturing because of her strategic location and lack of natural

resources.

Factor 2 is driven by a different set of variables, most notably by variables that are

closely linked to the labor market, as well as the business expectations on the performance of

Singapore’s major industries. Lastly, the second factor loads the variables related to the number

of companies formed in Singapore. The 3rd

and 4th

factors contribute less to the total variance

explained, and can be broadly described as factors pertaining to Singapore’s financial indicators

and company formation respectively. Again, we take caution in interpreting the common factors,

but further research might allow for some insight into not only what indicators really drive these

common factors, as well as policy implications, if any, that can be derived from the interpretation

of these common factors.

7.2 Empirical Results

The results of the forecasting experiment are listed in Table 1-5, where we evaluate the

performance of each forecasting model by the Root Mean Squared Error (RMSE) defined as:

∑

(22)

where

are the observed values of the h period ahead growth rates,

are the

estimated forecasts of growth rates, and T is the number of periods being forecasted in the

exercise. Appendix B and C shows the RMSE for the AO random walk model and relative

performances of the other forecasting models. The entries in the appendix are the RMSE of the

candidate forecasting model as a ratio of the RMSE of the benchmark forecasting model. A ratio

of less than 1 signifies more accurate forecasts, while a ratio of more than 1 would mean that the

model in question did not outperform a simple random walk. Appendix B shows the results from

a fixed window inflation forecasting exercise, and Appendix C shows the results of utilizing a

rolling window in the inflation forecasting exercise. Appendix D and E shows the results for IPI

growth forecasting for the fixed and rolling window estimates respectively.

7.2.1 Fixed vs Rolling Window Forecasts

Economic forecasting literature have recognized the trade-offs between using a fixed or

rolling window in data selection. Clark and McCracken (2004) explained that a rolling window

forecast would allow researchers to avoid the biases associated with structural change, but the

smaller amount of observable data would mean a larger variance in parameter estimates.

For inflation forecasts using both a fixed window as well as a rolling window, we find

that for factor forecasting models, the rolling window estimates generally outperformed the fixed

window estimates at the 3 and 4 quarter ahead forecasts. For the rest of the models, the fixed

window inflation forecast estimates generally outperformed the rolling window estimates at all

forecasting horizons.

For the case of IPI growth, the results were slightly different. For the AR models, fixed

window forecasts outperformed rolling window forecasts. For the factor models with lags, the

fixed window forecasts outperformed almost all of the rolling window forecasts. For factor

models without lags, the rolling window forecasts performed better in the long range horizons of

3, 4, and 8 step ahead forecasts. In summary, it is difficult to pinpoint an exact relationship

between the forecast performance using either estimation windows for all the models, which

reflects the ambiguity and trade-offs from using both methods.

7.2.2 Evaluation of the Forecasting Models

Firstly, we outline the general results we find from both rolling and fixed window

forecasting exercises. At the 1 period ahead horizon, the AO random walk model outperformed

all our forecasting models. For the inflation forecasts, at the 3, 4 and 8 quarters ahead horizons,

our factor forecasting models generally outperformed the AO random walk for both the fixed and

rolling window procedures. For the IPI growth forecasts, the only improvement was seen at the 8

quarter ahead horizon, the improvement at this horizon was more than the improvement for

inflation forecasting. A more significant finding would be that the inclusion of lags in the factor

models actually caused the performance of the factor model to deteriorate significantly compared

to the factor-only model without any lags.

7.2.2.1 Inflation Forecasting

We found mixed results in performance of factor models for inflation forecasting and IPI

growth forecasting. In the case of inflation forecasting, we see that under fixed window

estimation, the factor model without lags that used one factor produced the best overall

performance. This result is surprising because the first factor only accounts for 17.27% of total

variance explained in the dataset, and we would expect using more factors to improve forecast

performance. Under the rolling window procedure, the factor-only model that used the Bai-Ng

test to select the optimal number of factors showed the most accurate forecasts. This result is the

same as Stock and Watson (1999, 2002) that factor models do indeed improve inflation forecast

accuracy.

As for the competing models, we find that the AR model also outperformed the AO

random walk model at the 4 and 8 quarter ahead horizons. However, for the MLI model and

Phillips Curve model, it seems that the ability to consistently make accurate forecasts broke

down. They outperform the AO random walk model at the 3 and 4 quarter ahead forecast when

using the fixed window estimation, and outperform the AO random walk model only at the 8

quarter ahead forecast when using the rolling window estimation. It is interesting to analyze if

the Phillips Curve and MLI model still has predictive abilities in the Singaporean context, and

warrants further research.

7.2.2.2 IPI Growth Forecasting

For IPI growth forecasting, the factor-only models produced the best forecasts amongst

the factor augmented forecasting models, but surprisingly did not show a significant

improvement from the AR model. As mentioned in a previous section, the inclusion of lags

caused forecasting performance to deteriorate, and in the case of IPI growth forecasting, the

factor models with lags performed worse than the AR model.

7.2.3 Selection of the Optimal Number of Factors and Lags

In recent literature with regards to macroeconomic forecasting, many papers utilized

information criterions such as the BIC, AIC and the Bai-Ng test to determine the optimal number

of lags and factors in their dynamic factor forecasting models. In this paper, we ran forecasting

models that utilized these criterions, as well as ran a range of models of all the possible

combinations of lags and factors. Thus it would be insightful to see if these criterions actually

improve forecasts for Singapore, for both factor models as well as the other models utilized in

this paper.

We find that for the inflation forecasting model with factors and lags of inflation using

the fixed window procedure, using the Bai-Ng test to select the optimal number of factors and

then the BIC to select the optimal number of lags resulted in a significant improvement in

forecasting performance especially in the 1, 2, 4 and 8 quarter ahead horizons. Under the rolling

window procedure, the utilization of the Bai-Ng test in the factors-only model improved upon

the inflation forecasting performances of the fixed-factors-only model.

However, in the rest of the models that used the Bai-Ng test to forecast inflation or IPI

growth, the implementation of the Bai-Ng test did not result in any conclusive improvement in

forecasting ability. We see that the implementation of the Bai-Ng test gave mixed results, and

further research into different selection techniques should be performed to see which selection

model would improve upon the fixed-factor forecast model.

7.2.4 Statistical Significance of Improvements in Forecasting Ability

The numbers in parenthesis below the ratio of RMSE are the p-value from the Diebold-

Mariano or Giancomini-White test statistic. We see that when the inflation and IPI growth

forecast models are compared to the AO Random Walk model, in the cases where we see an

improvement in forecast performance, it is generally significant only at the 8 quarter ahead

horizon where there is great improvement in forecasting performance. This is the case for the AR

models, Phillips curve model, and some of the variations of the models that include common

factors in the forecast model.

In section 7.2.2.2, we showed that there was no obvious improvement in IPI growth

forecasting performance when we included factors in the forecasting models. We switch our

focus towards inflation forecasting models, and to better evaluate their effectiveness, we

compare the factor model forecasts not only to the AO random walk model, but to the other

models which have shown the best performances.

Table 2. Comparisons of 4 Quarter Ahead Forecasts for Inflation

AO

Random

Walk

MLI

Forecast

Phillips

Curve

AR

Model,

Lags

chosen by

BIC

Factor

Selection

using Bai-

Ng

Lag

Selection

using BIC

and Factor

Selection

using Bai-

Ng

AO Random - 0.606023 1.172285 1.208294 1.381087 1.303677

Walk (0.176549) (0.542642) (0.521506) (0.426565) (0.475526)

MLI Forecast

1.650102

(0.176549) -

1.93439

(0.062892)

1.993808

(0.062425)

2.278935

(0.068546)

2.151201

(0.069162)

Phillips Curve

0.853035

(0.542642)

0.516959

(0.062892) -

1.030717

(0.095048)

1.178116

(0.165976)

1.112083

(0.630646)

AR Model,

Lags chosen by

BIC

0.827613

(0.521506)

0.501553

(0.062425)

0.970198

(0.095048) -

1.143006

(0.333762)

1.078941

(0.761078)

Factor Selection

using Bai-Ng

0.724067

(0.426565)

0.438801

(0.068546)

0.848813

(0.165976)

0.874886

(0.333762) -

0.94395

(0.212215)

Lag Selection

using BIC and

Factor Selection

using Bai-Ng

0.767061

(0.475526)

0.464857

(0.069162)

0.899214

(0.630646)

0.926835

(0.761078)

1.059378

(0.212215) -

Table 2 is a tabular comparison of two versions of the factor forecasting models as well

as the AO random walk model, MLI model, Phillips Curve model, and the AR model. We only

make comparisons of the 4 quarter ahead forecast, and we only consider the results in the lower

diagonal of the symmetric 6x6 table. The numbers in the first row of each cell shows the ratio of

the Root Mean Squared Errors (RMSE of the row model/RMSE of the column model. From the

table, we see that the factor models do indeed outperform all the other models, but the

improvements from using the factor models are not statistically significant at the 10%

significance level, with the exception of the MLI model. Thus we see that even though factor

forecasting models do outperform their peers under the RMSE comparison metric, we should

take caution in accepting these results from the outset because the improvements in their

forecasting ability is not statistically significant.

8. Conclusions

In this study, we examine the out-of-sample forecasting performance of factor models in the

case of Singapore for the period from 1992:Q2 to 2009 Q4. We described the variables gathered,

the models used, the methodology to achieve the forecasts, and the forecast evaluation

techniques, to ultimately determine if the inclusion of factor analysis could improve upon

conventional forecasting models. Comparing the results of the factor forecasting model, we

reached three main conclusions.

Firstly, all the factor models we used in the exercise outperformed the AO Random Walk

model only at the longer forecast horizons. Only in the case of inflation forecasting, did factor

models outperform all the other models, signifying increased accuracy. In the case of IPI growth

forecasting, factor models did not contribute to any improvement in forecasting ability. Secondly,

the inclusion of the Bai-Ng test to select the optimal number of factors did not conclusively

improve forecast performance in all our forecasting exercises. And lastly, even though the use of

factor models did indeed result in lower RMSE in forecasting performance when compared to

the random walk model, the difference is only significant at the 8 quarters ahead horizon. Thus

there is a possibility that the improvement in performance when using factor models could be

attributed to chance, and this problem can be resolved with a greater number of observations.

We conclude that there is some evidence that factor models are useful for inflation but not for

IPI growth forecasting purposes in the case of Singapore. More analysis should be done to

determine why the performances of factor driven inflation forecasting models differ across the

variables to be forecasted. One hypothesis could be that the factor model does not perform as

well when the variable to be forecasted is extremely volatile. Further improvements could be

made to the factor models by involving factor analysis in non-parametric models, the use of a

larger more comprehensive data set, or a more effective factor selection technique, which might

improve the performance of factor models and become a reliable tool for policymakers around

the world.

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Appendix A

Description Mnemonic Tran

1) Real GDP

1 GROSS DOMESTIC PRODUCT AT CURRENT

MARKET PRICES

GDPMKTPRICE ∆ln

2 EXPENDITURE ON GDP EXPGDP ∆ln

3 PRIVATE CONSUMPTION EXPENDITURE PVTCONEXP ∆ln

4 GOVERNMENT CONSUMPTION EXPENDITURE GOVCONEXP ∆ln

5 GROSS FIXED CAPITAL FORMATION FXCAPFORM ∆ln

6 GROSS FIXED CAPITAL FORMATION

CONSTRUCTION & WORKS

FXCAPFORMCONSTRUCT ∆ln

7 GROSS FIXED CAPITAL FORMATION

RESIDENTIAL BUILDINGS

FXCAPFORMRESBUILD ∆ln

8 GROSS FIXED CAPITAL FORMATION NON-

RESIDENTIAL BUILDINGS

FXCAPFORMNONRESBUILD ∆ln

9 GROSS FIXED CAPITAL FORMATION OTHER

CONSTRUCTION

FXCAPFORMOTHERCONSTRUCT ∆ln

2) Trade Indicators

10 EXPORTS OF GOODS AND SERVICES EXGDAS ∆ln

11 IMPORTS OF GOODS AND SERVICES IMGDAS ∆ln

12 EXTERNAL TRADE: TOTAL EXTRADETOTAL ∆ln

13 EXTERNAL TRADE: TOTAL OIL EXTRADETOTALOIL ∆ln

14 EXTERNAL TRADE: TOTAL IMPORTS EXTRADETOTALM ∆ln

15 EXTERNAL TRADE: TOTAL IMPORTS OIL EXTRADEOILM ∆ln

16 EXTERNAL TRADE: IMPORTS NON OIL EXTRADENONOILM ∆ln

17 EXTERNAL TRADE: TOTAL EXPORTS EXTRADETOTALX ∆ln

18 EXTERNAL TRADE: OIL EXPORT EXTRADEOILX ∆ln

19 EXTERNAL TRADE: NON OIL EXPORT EXTRADENONOILX ∆ln

20 EXTERNAL TRADE: TOTAL REEXPORTS EXTRADERE ∆ln

21 EXTERNAL TRADE: REXPORT OIL EXTRADEREOIL ∆ln

22 EXTERNAL TRADE: REXPORT NON OIL EXTRADERENONOIL ∆ln

3) Foreign Exchange Rate

23 US DOLLAR USD ∆ln

24 STERLING POUND STER ∆ln

25 JAPANESE YEN JPY ∆ln

26 MALAYSIAN RINGGIT MALAYRM ∆ln

27 HONG KONG DOLLAR HKD ∆ln

28 KOREAN WON WON ∆ln

29 NEW TAIWAN DOLLAR NTD ∆ln

30 INDONESIAN RUPIAH INDO ∆ln

31 THAI BAHT BAHT ∆ln

4) Price Indices

32 GROSS DOMESTIC PRODUCT DEFLATORS AT

MARKET PRICE

GDPDEFFLATORMKTPX ∆2 ln

33 GROSS DOMESTIC PRODUCT DEFLATORS

MANUFACTURING

GDPDEFFLATORMANU ∆2 ln

34 GROSS DOMESTIC PRODUCT DEFLATORS

CONSTRUCTION

GDPDEFFLATORCONSTRUCT ∆2 ln

35 GROSS DOMESTIC PRODUCT DEFLATORS

SERVICE

GDPDEFFLATORSERVICE ∆2 ln

36 DOMESTIC SUPPLY PRICE INDEX DOMSUPPI ∆2 ln

37 IMPORT PRICE INDEX: OVERALL MPIOVERALL ∆2 ln

38 IMPORT PRICE INDEX: NONOIL MPINONOIL ∆2 ln

39 EXPORT PRICE INDEX: OVERALL XPIOVERALL ∆2 ln

40 EXPORT PRICE INDEX: NONOIL XPINONOIL ∆2 ln

41 CPI: ALL ITEMS CPIALL ∆2 ln

42 CPI: FOOD CPIFOOD ∆2 ln

43 CPI: CLOTHING CPICLOTHES ∆2 ln

44 CPI: TRANSPORT CPITRAN ∆2 ln

45 CPI: HEALTH CPIHEALTH ∆2 ln

46 CPI: HOUSEHOLD ITEMS CPIHHG ∆2 ln

47 CPI: NONDURABLE HOUSEHOLD GOODS CPINDHHG ∆2 ln

5) Sectorial Indicators

48 FORMATION OF COMPANIES: REAL ESTATE,

RENTAL AND LEASING ACTIVITIES

FORMRE ∆ln

49 FORMATION OF COMPANIES: TOTAL FORMTOTAL ∆ln

50 FORMATION OF COMPANIES:

MANUFACTURING

FORMMAN ∆ln

51 FORMATION OF COMPANIES: CONSTRUCTION FORMCONSTR ∆ln

52 FORMATION OF COMPANIES: WHOLESALE

AND RETAIL COMPANIES

FORMWRC ∆ln

53 FORMATION OF COMPANIES: TRANSPORT

AND STORAGE

FORMTRANS ∆ln

54 FORMATION OF COMPANIES: HOTELS AND

RESTAURANTS

FORMHOTEL ∆ln

55 FORMATION OF COMPANIES: INFO AND

COMM

FORMINFO ∆ln

56 FORMATION OF COMPANIES: FINANCIAL AND

INSURANCE

FORMFIN ∆ln

57 VISTOR ARRIVAL VISTOR ∆ln

58 RETAIL SALES INDEX RETAILSI ∆ln

59 AIR CARGO DISCHARGED AIRCARGO ∆ln

60 NEW REGISTRATION OF MOTOR VEHICLES MOTOR ∆ln

61 SEA PASSENGERS SEAPASSENGERS ∆ln

62 COMPOSITE LEADING INDEX COMLI ∆ln

63 ELECTRICITY GENERATION AND SALES ELECTRICITY ∆ln

6) Labor Market

64 RETRENCHED WORKERS RETRENCH ∆ln

65 UNIT LABOUR COST INDEX LABCOSTS ∆ln

66 UNIT BUSINESS COST INDEX OF

MANUFACTURING: OVERALL

BIZCOSTOVERALL ∆ln

67 UNIT BUSINESS COST INDEX OF

MANUFACTURING: MANUFACTURING

BIZCOSTMANU ∆ln

68 EMPLOYMENT BY SECTOR: TOTAL EMPLOYALL ∆ln

69 UNEMPLOYMENT RATE: TOTAL UNEMPRATE ∆level

70 RESIDENT UNEMPLOYMENT RATE UNEMPRES ∆level

71 AVERAGE MONTHLY EARNINGS: TOTAL WAGETOTAL ∆2 ln

72 EMPLOYMENT CHANGES: SERVICES SECTOR

TOTAL EMPLOYMENT

EMPCHANGETOTAL Level

73 EMPLOYMENT CHANGES: SERVICES SECTOR

WHOLESALE AND RETAIL TRADE

EMPLOYMENT

EMPCHANGEWRT Level

74 EMPLOYMENT CHANGES: HOTELS AND

CATERING EMPLOYMENT

EMPCHANGEHOTEL Level

75 EMPLOYMENT CHANGES: TRANSPORT AND

STORAGE EMPLOYMENT

EMPCHANGETRANS Level

76 EMPLOYMENT CHANGES: FINANCIAL

SERVICES EMPLOYMENT

EMPCHANGEFIN Level

77 EMPLOYMENT CHANGES: REAL ESTATE

EMPLOYMENT

EMPCHANGERE Level

78 EMPLOYMENT CHANGES: BUSINESS EMPCHAGNEBIZ Level

SERVICES EMPLOYMENT

7) Construction

79 HDB RESALE PRICE INDEX HDBPX ∆ln

80 PROPERTY PRICE INDEX: RESIDENTIAL PROPPIRES ∆ln

81 PROPERTY PRICE INDEX: OFFICE PROPPIOFFICE ∆ln

82 PROPERTY PRICE INDEX: SHOP SPACE PROPPISHOP ∆ln

83 VALUE OF CONTRACTS AWARDED BY

SECTOR AND TYPE OF WORK: TOTAL PUBLIC

& PRIVATE SECTOR

CONTRACTTOTAL ∆ln

84 VALUE OF CONTRACTS AWARDED BY

SECTOR AND TYPE OF WORK: PUBLIC

SECTOR

CONTRACTPUBLIC ∆ln

85 VALUE OF CONTRACTS AWARDED BY

SECTOR AND TYPE OF WORK: PRIVATE

SECTOR

CONTRACTPRIVATE ∆ln

86 VALUE OF CONTRACTS AWARDED BY

SECTOR AND TYPE OF WORK: RESIDENTIAL

BUILDING

CONTRACTRES ∆ln

87 VALUE OF CONTRACTS AWARDED BY

SECTOR AND TYPE OF WORK: COMMERCIAL

BUILDING

CONTRACTCOMM ∆ln

88 VALUE OF CONTRACTS AWARDED BY

SECTOR AND TYPE OF WORK: INDUSTRIAL

BUILDING

CONTRACTINDUS ∆ln

89 VALUE OF CONTRACTS AWARDED BY

SECTOR AND TYPE OF WORK: CIVIL

ENGINEERING

CONTRACTCIVIL ∆ln

90 VALUE OF CONTRACTS AWARDED BY

SECTOR AND TYPE OF WORK:

INSTITUTIONAL

CONTRACTINSTITUTE ∆ln

8) Industrial Production

91 INDEX OF INDUSTRIAL PRODUCTION: TOTAL IIPTOTAL ∆ln

92 INDEX OF INDUSTRIAL PRODUCTION:

PETROLEUM

IIPPETRO ∆ln

93 INDEX OF INDUSTRIAL PRODUCTION:

CHEMICAL

IIPCHEM ∆ln

94 INDEX OF INDUSTRIAL PRODUCTION:

PHARMACEUTICALS

IIPPHARM ∆ln

95 INDEX OF INDUSTRIAL PRODUCTION:

MACHINERY

IIPMACHINE ∆ln

96 INDEX OF INDUSTRIAL PRODUCTION:

ELECTRICAL MACHINERY

IIPELECMACH ∆ln

97 INDEX OF INDUSTRIAL PRODUCTION:

ELECTRONICS

IIPELECTRONIC ∆ln

98 INDEX OF INDUSTRIAL PRODUCTION:

TRANSPORT

IIPTRANS ∆ln

99 INDEX OF INDUSTRIAL PRODUCTION:

FURNITURE & MANUFACTURING

IIPFURN ∆ln

9) Monetary Indicators

100 MONEY SUPPY: M3 MTHREE ∆2 ln

101 MONEY SUPPY: M2 MTWO ∆2 ln

102 MONEY SUPPY: M1 MONE ∆2 ln

103 MONEY SUPPY: CURRENCY CURRENCY ∆2 ln

104 MONEY SUPPY: DEMAND DEPOSITS DEMANDD ∆2 ln

105 MONEY SUPPY: FIXED DEPOSITS FIXEDD ∆2 ln

106 BANK LOANS: TOTAL BANKTOTAL ∆2 ln

107 BANK LOANS: MANUFACTURING BANKMANU ∆2 ln

108 BANK LOANS: BUILDING BANKBUILD ∆2 ln

109 BANK LOANS: GENERAL COMMERCE BANKCOMM ∆2 ln

110 BANK LOANS: FINANCIAL INSTITUTIONS BANKFIN ∆2 ln

111 BANK LOANS: PROFESSIONAL AND PRIVATE

INDIVIDUALS

BANKINDIV ∆2 ln

10) Indicators

112 PRIME LENDING RATE PRIMELENDING Level

113 3-MONTH INTERBANK RATE THREEIBR Level

114 3-MONTH US$ SIBOR THREEUS Level

115 GOVERNMENT SECURITIES - 3-MONTH

TREASURY BILLS YIELD

THREEMONTHTBILL Level

116 GOVERNMENT SECURITIES - 5-YEAR BOND

YIELD

FIVEYEARBOND Level

117 GOVERNMENT SECURITIES - 2-YEAR BOND

YIELD

TWOYEARBOND Level

118 3-MONTH COMMERCIAL BILLS THREEMONTHCBILL Level

119 GOVERNMENT SECURITIES - 1-YEAR

TREASURY BILLS YIELD

ONEYEARTBILL Level

11) Business Expectations:

120 BUSINESS EXPECTATIONS OF THE

MANUFACTURING SECTOR: GENERAL

BUSINESS EXPECTATIONS (FORECAST FOR

NEXT 6 MONTHS)

BIZEXPGEN Level

121 BIZ EXP: TOTAL NEW ORDERS RECEIVED

(FORECAST FOR NEXT QUARTER)

BIZNEWORDER Level

122 BIZ EXP: EXPORT ORDERS (FORECAST FOR

NEXT QUARTER)

BIZEXPEXPORT Level

123 BIZ EXP: DELIVERIES IN SINGAPORE

(FORECAST FOR NEXT QUARTER)

BIZEXPDEL Level

124 BIZ EXP: DELIVERIES OVERSEAS (FORECAST

FOR NEXT QUARTER)

BIZEXPDELO Level

125 BIZ EXP: WHOLESALE & RETAIL TRADE

ENDING STOCKS FORECAST

BIZEXPWRTENDSTOCKFOR Level

126 BIZ EXP: WHOLESALE & RETAIL TRADE

ENDING STOCKS PERFORMANCE

BIZEXPWRTENDSTOCKPER Level

127 BUSINESS EXPECTATIONS FOR THE SERVICES BIZEXPHOTELSENDSTOCK Level

SECTOR (PERFORMANCE) - ENDING STOCKS

OF MERCHANDISE, BIZ EXP: HOTELS &

CATERING

128 BUSINESS EXPECTATIONS FOR THE SERVICES

SECTOR (FORECAST FOR THE NEXT

QUARTER) - GENERAL BUSINESS

BIZEXPSERTOTALFOR Level

129 BIZ EXP: TOTAL SERVICES SECTOR

(PERFORMANCE)

BIZEXPTOTALPER Level

130 BIZ EXP: WHOLESALE & RETAIL TRADE

(PERFORMANCE)

BIZEXPWRTPER Level

131 BIZ EXP: HOTELS & CATERING

(PERFORMANCE)

BIZEXPHOTELPER Level

132 BIZ EXP: TRANSPORT & STORAGE

(PERFORMANCE)

BIZEXPTRANSPER Level

133 BIZ EXP: FINANCIAL SERVICES

(PERFORMANCE)

BIZEXPFINPER Level

134 BIZ EXP: BANKS & FINANCE COMPANIES

(PERFORMANCE)

BIZEXPBANKSPER Level

135 BIZ EXP: REAL ESTATE (PERFORMANCE) BIZEXPREPER Level

136 BIZ EXP: BUSINESS SERVICES

(PERFORMANCE)

BIZEXPBIZPER Level

12) FOREIGN COMPOSITE LEADING INDEX

137 COMPOSITE LEADING INDEX GERMANY COMPIGERMANY ∆ln

138 COMPOSITE LEADING INDEX JAPAN COMPIJAPAN ∆ln

139 COMPOSITE LEADING INDEX UNITED

KINGDOM

COMPIUK ∆ln

140 COMPOSITE LEADING INDEX UNITED STATES COMPIUS ∆ln

141 COMPOSITE LEADING INDEX EURO AREA COMPIEURO ∆ln

142 COMPOSITE LEADING INDEX FOUR BIG

EUROPEAN

COMPIBIGFOUREURO ∆ln

143 COMPOSITE LEADING INDEX KOREA COMPIKOREA ∆ln

144 COMPOSITE LEADING INDEX MAJOR FIVE

ASIA

COMPIASIA ∆ln

13) FOREIGN STOCK PRICES

145 STOCK PRICE: AUSTRALIA STOCKAUS ∆ln

146 STOCK PRICE: JAPAN STOCKJAP ∆ln

147 STOCK PRICE: EURO AREA STOCKEURO ∆ln

148 STOCK PRICE: KOREA STOCKKOREA ∆ln

149 STOCK PRICE: UNITED KINGDOM STOCKUK ∆ln

150 STOCK PRICE: UNITED STATES STOCKUS ∆ln

14) FOREIGN GDP

151 QUARTERLY GROWTH RATES OF GDP:

AUSTRALIA

AUSGDP Level

152 QUARTERLY GROWTH RATES OF GDP: JAPAN JAPGDP Level

153 QUARTERLY GROWTH RATES OF GDP: KOREA KOREAGDP Level

154 QUARTERLY GROWTH RATES OF GDP:

UNITED KINGDOM

UKGDP Level

155 QUARTERLY GROWTH RATES OF GDP: USA USAGDP Level

Appendix B Inflation Forecast Using Fixed Window

Table 1

AO Random-Walk Model Number of quarters ahead

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

RMSE 0.020156 0.025432 0.028614 0.029474 0.024249

Table 2

Direct Autoregressive Model Number of periods ahead

Number of

lags

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

1 1.437441 1.082763 0.861052 0.746242 0.662707

(0.04752) (0.34619) (0.21916) (0.06851) (0.06391)

2 1.511935 1.128133 0.869112 0.735881 0.664973

(0.03841) (0.26439) (0.21279) (0.07523) (0.06033)

3 1.573489 1.166454 0.863166 0.73078 0.665615

(0.0305) (0.19821) (0.20401) (0.07886) (0.05652)

4 1.604399 1.170219 0.852184 0.739614 0.66133

(0.02255) (0.182) (0.20205) (0.08309) (0.04943)

5 1.596307 1.170251 0.839326 0.736606 0.66841

(0.01945) (0.15423) (0.20017) (0.08749) (0.04617)

BIC 1.437441 1.082763 0.861052 0.746242 0.670956

(0.04752) (0.34619) (0.21916) (0.06851) (0.03491)

Table 3

Multivariate Leading Index

Model

Number of periods ahead

Number of

lags

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

BIC 1.466259 1.124545 0.902328 0.787422 1.198649

(0.04722) (0.27222) (0.31621) (0.09596) (0.3543)

Table 4

Autoregressive Phillips Curve

Model

Number of periods ahead

Number of

lags

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

BIC 1.462022 1.131981 0.91325 0.792831 1.018858

(0.04542) (0.26828) (0.33404) (0.09502) (0.47926)

Table 5

Factor Model Version (i) Number of periods ahead

Number of

lags

Number of

Factors

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

0 Bai-Ng

Test 1.353218 0.962655 0.815328 0.714488 0.632438

(0.08593) (0.44101) (0.21803) (0.0999) (0.0657)

Table 6

Factor Model Version (ii) Number of periods ahead

Number of

lags

Number of

Factors

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

BIC Bai-Ng

Test 1.332313 1.047948 0.927169 0.75452 0.670007

(0.09485) (0.41943) (0.38542) (0.10174) (0.05395)

Table 7

Factor Model Version (iii) Number of periods ahead

No Lags Number of

Factors

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

1 1.364057 0.949791 0.784691 0.702258 0.628475

(0.07493) (0.42709) (0.20041) (0.09864) (0.06468)

2 1.351544 0.968871 0.816181 0.711611 0.630555

(0.08682) (0.45089) (0.21966) (0.09808) (0.06588)

3 1.355728 0.959205 0.817838 0.712501 0.62742

(0.08665) (0.43544) (0.21916) (0.09851) (0.06571)

4 1.38229 0.952813 0.844978 0.722386 0.63507

(0.0834) (0.42295) (0.24171) (0.09251) (0.06311)

5 1.371731 0.934652 0.849274 0.724605 0.64294

(0.08475) (0.38994) (0.23599) (0.08913) (0.06847)

6 1.38112 0.943741 0.842564 0.724655 0.641038

(0.0865) (0.40159) (0.22392) (0.0881) (0.06599)

7 1.387655 0.9426 0.852948 0.736552 0.636935

(0.07776) (0.39855) (0.23029) (0.08388) (0.07675)

8 1.393868 0.944583 0.845972 0.740791 0.656799

(0.07454) (0.40382) (0.23166) (0.08057) (0.0808)

9 1.391421 0.957341 0.851686 0.738103 0.644018

(0.07659) (0.42332) (0.24309) (0.08311) (0.09195)

10 1.377824 0.961075 0.852893 0.747824 0.653063

(0.07531) (0.43067) (0.24705) (0.08775) (0.09596)

11 1.351278 0.957908 0.86032 0.747148 0.663278

(0.10219) (0.42114) (0.24529) (0.08932) (0.08436)

12 1.343455 0.96491 0.854101 0.746076 0.655289

(0.11024) (0.43301) (0.23092) (0.08162) (0.07591)

Table 8

Factor Model Version (iv) Number of periods ahead

Number of

lags

Number of

Factors

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

1 1 1.816758 1.321669 1.03525 0.881352 0.692457

(0.01255) (0.06708) (0.41109) (0.07732) (0.04498)

2 1 1.37487 1.071865 0.85128 0.72722 0.641931

(0.08773) (0.38627) (0.26485) (0.10238) (0.06193)

3 1 1.407145 1.097433 0.837657 0.734639 0.637413

(0.07392) (0.3497) (0.2483) (0.10361) (0.05221)

4 1 1.432188 1.115573 0.842646 0.749476 0.642086

(0.05531) (0.32442) (0.25687) (0.10503) (0.04976)

5 1 1.461427 1.130405 0.864411 0.77149 0.650059

(0.04131) (0.30608) (0.2807) (0.10612) (0.05139)

1 2 1.853861 1.32069 1.043773 0.89913 0.693695

(0.01066) (0.07655) (0.3901) (0.07428) (0.04649)

2 2 1.380643 1.113997 0.894158 0.75998 0.641027

(0.07929) (0.32912) (0.33056) (0.10196) (0.06138)

3 2 1.414435 1.149465 0.901097 0.794862 0.632801

(0.07189) (0.29527) (0.34134) (0.09251) (0.05244)

4 2 1.449473 1.176981 0.912222 0.826268 0.640971

(0.05405) (0.26829) (0.35632) (0.08959) (0.05024)

1 3 1.853777 1.31527 1.046354 0.901715 0.697158

(0.0115) (0.0794) (0.38177) (0.07469) (0.0456)

2 3 1.383693 1.100321 0.888783 0.76218 0.639413

(0.08135) (0.34746) (0.3236) (0.10302) (0.05677)

3 3 1.413853 1.147675 0.900301 0.790072 0.637616

(0.07087) (0.29836) (0.33945) (0.09319) (0.05669)

4 3 1.446208 1.167081 0.922494 0.84843 0.634896

(0.05492) (0.28039) (0.37214) (0.07466) (0.04849)

5 3 1.588829 1.25499 0.965724 0.905513 0.649239

(0.02086) (0.23298) (0.45297) (0.09027) (0.05072)

1 4 1.820791 1.338801 1.089253 0.930433 0.7156

(0.00783) (0.06563) (0.23273) (0.07226) (0.04129)

2 4 1.338157 1.114636 0.92399 0.795775 0.67177

(0.10537) (0.324) (0.3727) (0.08677) (0.06626)

3 4 1.370704 1.167633 0.944607 0.818331 0.641991

(0.09791) (0.27866) (0.40182) (0.08428) (0.05594)

4 4 1.416726 1.18356 0.940528 0.850933 0.639892

(0.07937) (0.24556) (0.39937) (0.07387) (0.04956)

5 4 1.608559 1.242392 0.983038 0.927239 0.650588

(0.01547) (0.234) (0.47679) (0.08873) (0.05452)

1 5 1.822867 1.329217 1.10606 0.929592 0.717666

(0.00792) (0.06395) (0.13993) (0.06922) (0.0475)

2 5 1.340349 1.085953 0.915468 0.785171 0.663785

(0.10241) (0.3619) (0.34809) (0.088) (0.06907)

3 5 1.37766 1.145702 0.936494 0.811703 0.650711

(0.09278) (0.30233) (0.37699) (0.08048) (0.0589)

4 5 1.42335 1.14657 0.908064 0.824059 0.635766

(0.07386) (0.286) (0.34148) (0.07274) (0.05166)

5 5 1.621096 1.264109 1.002998 0.883927 0.678098

(0.01525) (0.22188) (0.49586) (0.09179) (0.08837)

1 6 1.819798 1.333948 1.095749 0.929668 0.724938

(0.00709) (0.05599) (0.15838) (0.0689) (0.04906)

2 6 1.372485 1.090525 0.903819 0.777229 0.654584

(0.08866) (0.35761) (0.32873) (0.08805) (0.06776)

3 6 1.389433 1.164811 0.930669 0.806332 0.641915

(0.07467) (0.27218) (0.3594) (0.07989) (0.056)

4 6 1.431628 1.167623 0.91085 0.825281 0.637084

(0.06247) (0.25327) (0.34341) (0.07122) (0.05416)

5 6 1.62635 1.272419 0.989551 0.879988 0.682607

(0.01405) (0.21202) (0.4851) (0.0914) (0.08873)

1 7 1.843806 1.344365 1.103709 0.969191 0.735371

(0.00876) (0.05493) (0.18629) (0.06651) (0.03691)

2 7 1.356286 1.113877 0.909172 0.779272 0.649878

(0.09326) (0.32509) (0.33769) (0.08288) (0.07041)

3 7 1.422158 1.170617 0.919829 0.804949 0.650682

(0.04842) (0.26643) (0.33835) (0.07667) (0.05568)

4 7 1.441339 1.184437 0.91485 0.829955 0.647308

(0.06298) (0.23332) (0.35267) (0.06582) (0.05748)

5 7 1.65652 1.26586 0.984452 0.884972 0.680331

(0.01282) (0.21273) (0.47778) (0.0822) (0.086)

1 8 1.834814 1.336682 1.105877 0.969938 0.731993

(0.00872) (0.06445) (0.18032) (0.06854) (0.0305)

2 8 1.3445 1.098259 0.911702 0.786354 0.639154

(0.08542) (0.35259) (0.3425) (0.08098) (0.05902)

3 8 1.409338 1.144625 0.918137 0.809399 0.652189

(0.04877) (0.30648) (0.33359) (0.07725) (0.05295)

4 8 1.441858 1.180588 0.913955 0.836534 0.66258

(0.04922) (0.24257) (0.35083) (0.06924) (0.06739)

5 8 1.639143 1.256499 0.986119 0.889444 0.699727

(0.00798) (0.21802) (0.48027) (0.08) (0.09575)

1 9 1.838499 1.353782 1.10468 0.96316 0.737998

(0.00692) (0.05558) (0.20069) (0.06709) (0.03045)

2 9 1.327646 1.087341 0.910584 0.796562 0.644654

(0.09164) (0.3699) (0.3432) (0.08191) (0.05365)

3 9 1.4222 1.166569 0.950085 0.832056 0.689286

(0.04185) (0.2914) (0.39639) (0.07023) (0.05223)

4 9 1.441614 1.20656 0.949253 0.892197 0.687536

(0.04712) (0.22489) (0.40962) (0.0538) (0.052)

5 9 1.638964 1.246824 1.005401 0.925538 0.703552

(0.00792) (0.23507) (0.49254) (0.07909) (0.09752)

1 10 1.845398 1.333139 1.110443 0.956677 0.742753

(0.0058) (0.05793) (0.20838) (0.06726) (0.03102)

2 10 1.326481 1.07038 0.918588 0.809331 0.651274

(0.09575) (0.38947) (0.35247) (0.08257) (0.06164)

3 10 1.411594 1.16055 0.956158 0.840316 0.682552

(0.04143) (0.29351) (0.40946) (0.06768) (0.04674)

4 10 1.414903 1.202318 0.960569 0.898064 0.688243

(0.0469) (0.22498) (0.42869) (0.05585) (0.05286)

5 10 1.513267 1.21218 0.996095 0.940171 0.714841

(0.01402) (0.25008) (0.49448) (0.0835) (0.10901)

1 11 1.829338 1.32992 1.096103 0.947961 0.761543

(0.00452) (0.05778) (0.20241) (0.05946) (0.04182)

2 11 1.26995 1.031442 0.887507 0.789943 0.669607

(0.12552) (0.44864) (0.28599) (0.0714) (0.06795)

3 11 1.385748 1.120656 0.929876 0.83002 0.713875

(0.04195) (0.33931) (0.36197) (0.07206) (0.06534)

4 11 1.402516 1.144073 0.946521 0.887472 0.674693

(0.04838) (0.27993) (0.40517) (0.0577) (0.05353)

5 11 1.522067 1.231976 0.99724 0.945555 0.763958

(0.01432) (0.23051) (0.4961) (0.08173) (0.15832)

1 12 1.807784 1.322345 1.086972 0.940399 0.756602

(0.00541) (0.06401) (0.22226) (0.06526) (0.03608)

2 12 1.269219 1.034278 0.890161 0.826379 0.679187

(0.10345) (0.44712) (0.29618) (0.06412) (0.07191)

3 12 1.412307 1.116139 0.922254 0.806452 0.701781

(0.04053) (0.34262) (0.35429) (0.06921) (0.072)

4 12 1.451825 1.144524 0.949093 0.866915 0.705013

(0.0427) (0.27669) (0.4125) (0.05939) (0.08974)

5 12 1.554862 1.218466 0.996754 0.953275 0.770572

(0.01532) (0.24158) (0.49542) (0.08195) (0.18869)

Appendix C Inflation Forecast Using Rolling Window

Table 1

AO Random-Walk Model Number of quarters ahead

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

RMSE 0.020156 0.025432 0.028614 0.029474 0.024249

Table 2

Direct Autoregressive Model Number of periods ahead

Number of

lags

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

1 1.504254 1.187336 0.985107 0.847962 0.722946

(0.0984) (0.4395) (0.9851) (0.4913) (0.054)

2 1.611495 1.362431 1.06142 0.877392 0.729724

(0.1082) (0.3251) (0.5786) (0.5449) (0.0683)

3 1.747693 1.465887 1.118506 0.925926 0.738741

(0.0714) (0.1106) (0.6619) (0.4257) (0.0824)

4 1.788609 1.454975 1.086826 0.923337 0.737677

(0.0382) (0.0996) (0.76) (0.6038) (0.0448)

5 1.926681 1.522186 1.083033 0.890774 0.716768

(0.0356) (0.0979) (0.5262) (0.5543) (0.0345)

BIC 1.509214 1.31289 0.9872 0.827613 0.731004

(0.094) (0.4205) (0.9812) (0.5215) (0.0371)

Table 3

Multivariate Leading Index

Model

Number of periods ahead

Number of

lags

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

BIC 2.409065 2.376254 2.373136 1.650102 0.886556

(0.0026) (0.0012) (0.0077) (0.1765) (0.4334)

Table 4

Autoregressive Phillips Curve

Model

Number of periods ahead

Number of

lags

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

BIC 1.506789 1.367609 1.018308 0.853035 0.757024

(0.105) (0.4066) (0.8778) (0.5426) (0.0832)

Table 5

Factor Model Version (i) Number of periods ahead

Number of

lags

Number of

Factors

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

0 Bai-Ng 1.419428 0.9621 0.780893 0.724067 0.705992

Test

(0.127) (0.8835) (0.5776) (0.4266) (0.0663)

Table 6

Factor Model Version (ii) Number of periods ahead

Number of

lags

Number of

Factors

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

BIC Bai-Ng

Test 1.68485 1.069491 0.811811 0.767061 0.863826

(0.101) (0.3326) (0.6081) (0.4755) (0.1324)

Table 7

Factor Model Version (iii) Number of periods ahead

No Lags Number of

Factors

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

1 1.404836 0.982712 0.814066 0.738956 0.69896

(0.0856) (0.9999) (0.6563) (0.4509) (0.0601)

2 1.423361 0.968681 0.786246 0.728238 0.745614

(0.1248) (0.9042) (0.5758) (0.4414) (0.1019)

3 1.406777 0.97838 0.79479 0.73346 0.748083

(0.1161) (0.9169) (0.5136) (0.4562) (0.111)

4 1.524566 1.037003 0.843927 0.7481 0.750203

(0.0603) (0.8827) (0.3722) (0.4778) (0.1011)

5 1.52282 1.048671 0.849994 0.750207 0.754543

(0.0837) (0.6388) (0.3537) (0.4827) (0.1209)

6 1.526981 1.046177 0.838813 0.750913 0.753073

(0.1146) (0.5977) (0.3485) (0.484) (0.1212)

7 1.578032 1.048308 0.841071 0.750523 0.762064

(0.0942) (0.6384) (0.3519) (0.4831) (0.1645)

8 1.625476 1.095425 0.851157 0.73652 0.763294

(0.0887) (0.6331) (0.3149) (0.4641) (0.173)

9 1.696446 1.103255 0.816431 0.73591 0.769487

(0.0588) (0.4619) (0.3329) (0.4476) (0.1848)

10 1.622881 1.049387 0.806861 0.737453 0.771141

(0.0632) (0.777) (0.334) (0.4572) (0.202)

11 1.633336 1.05375 0.797158 0.742469 0.773538

(0.0369) (0.7315) (0.3365) (0.4611) (0.1904)

12 1.673907 1.065819 0.790032 0.739957 0.770566

(0.0374) (0.7198) (0.3464) (0.464) (0.1968)

Table 8

Factor Model Version (iv) Number of periods ahead

Number of

lags

Number of

Factors

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

1 1 1.664288 1.0591634 0.7840825 0.7065539 0.7084216

(0.0788) (0.3539) (0.2627) (0.3893) (0.082)

2 1 1.6708711 1.3000977 0.8179281 0.7193894 0.7107106

(0.0908) (0.0303) (0.4643) (0.4052) (0.0847)

3 1 1.6624034 1.3711399 0.8393779 0.7339669 0.7201649

(0.0847) (0.0639) (0.4314) (0.4228) (0.0709)

4 1 1.5518997 1.3481308 0.9662618 0.7411814 0.7449118

(0.1692) (0.2428) (0.1704) (0.4224) (0.2162)

5 1 1.5244552 1.3590925 0.9402105 0.8279667 0.7477064

(0.2007) (0.2028) (0.1437) (0.5276) (0.1801)

1 2 1.6727029 1.0085036 0.7175171 0.683594 0.7046139

(0.0786) (0.4653) (0.4028) (0.3206) (0.0858)

2 2 1.6597415 1.274406 0.7755491 0.7201252 0.6896425

(0.1057) (0.0285) (0.589) (0.3684) (0.0961)

3 2 1.6152632 1.3467682 0.7973234 0.7479187 0.7329178

(0.0593) (0.0392) (0.6082) (0.3892) (0.064)

4 2 1.5777658 1.2656731 0.8533148 0.7149149 0.7522441

(0.1449) (0.1723) (0.2879) (0.3604) (0.1631)

1 3 1.6616267 1.013657 0.7213038 0.6808354 0.7033741

(0.087) (0.4922) (0.4167) (0.3305) (0.0871)

2 3 1.6512158 1.2603874 0.7764475 0.7250455 0.6959007

(0.1088) (0.0159) (0.567) (0.3751) (0.1184)

3 3 1.6117472 1.3090342 0.8049873 0.7566724 0.7196803

(0.0769) (0.0595) (0.6004) (0.3934) (0.0654)

4 3 1.5904654 1.2549119 0.8382993 0.725785 0.730016

(0.1577) (0.1587) (0.4268) (0.3753) (0.1113)

5 3 1.4724833 1.2414227 0.8195159 0.7873672 0.7497358

(0.2342) (0.1559) (0.5142) (0.4779) (0.0451)

1 4 1.6119523 1.031442 0.7721815 0.690988 0.7062401

(0.0885) (0.3609) (0.294) (0.3517) (0.0992)

2 4 1.6260352 1.2437603 0.7880434 0.7274655 0.70361

(0.122) (0.0449) (0.5763) (0.3992) (0.126)

3 4 1.6097403 1.2631681 0.8265464 0.7570313 0.7185116

(0.0855) (0.0872) (0.6273) (0.396) (0.0645)

4 4 1.5581422 1.2441897 0.8343237 0.7227245 0.7443359

(0.1717) (0.1726) (0.3763) (0.3766) (0.129)

5 4 1.4892932 1.2232594 0.8260941 0.8191448 0.7524759

(0.1984) (0.1666) (0.4268) (0.4827) (0.0842)

1 5 1.6176787 1.0223384 0.7660188 0.6848379 0.7122787

(0.0817) (0.3639) (0.2938) (0.3265) (0.1175)

2 5 1.6461132 1.2390004 0.7935509 0.7322707 0.7068637

(0.1096) (0.0451) (0.5846) (0.4223) (0.1487)

3 5 1.6338109 1.3109025 0.8236 0.7530897 0.744241

(0.064) (0.1301) (0.602) (0.3876) (0.2002)

4 5 1.5817668 1.2770376 0.8286575 0.7106038 0.7659614

(0.1444) (0.2469) (0.362) (0.3578) (0.2484)

5 5 1.5799188 1.2111757 0.8396522 0.8091232 0.7450806

(0.1049) (0.1428) (0.2469) (0.4636) (0.0366)

1 6 1.626887 1.0385314 0.7638195 0.6844855 0.7118859

(0.0658) (0.3282) (0.2818) (0.3232) (0.1144)

2 6 1.6494443 1.2336846 0.787 0.7376929 0.7137075

(0.1122) (0.0449) (0.5889) (0.4482) (0.1655)

3 6 1.6341951 1.3348846 0.8254147 0.7602153 0.7487247

(0.0715) (0.1132) (0.5829) (0.4) (0.1828)

4 6 1.586449 1.2944178 0.8347329 0.722487 0.7695747

(0.1238) (0.2365) (0.4259) (0.3813) (0.229)

5 6 1.6017094 1.2132155 0.8547078 0.79716 0.7317243

(0.0898) (0.1458) (0.1621) (0.481) (0.0449)

1 7 1.6233038 1.0478504 0.7544493 0.6901983 0.7073024

(0.073) (0.3366) (0.3021) (0.3459) (0.1284)

2 7 1.65283 1.2357587 0.7790967 0.7588737 0.7051132

(0.11) (0.0433) (0.5525) (0.4518) (0.1713)

3 7 1.6364275 1.3683469 0.848946 0.7892238 0.7332086

(0.074) (0.1476) (0.5405) (0.4228) (0.179)

4 7 1.5992606 1.2812232 0.8419107 0.7282023 0.7856665

(0.121) (0.2138) (0.5021) (0.3982) (0.2366)

5 7 1.619894 1.2751191 0.8533011 0.7960924 0.7332989

(0.0684) (0.1304) (0.1854) (0.4883) (0.0208)

1 8 1.6550269 1.0422835 0.7290622 0.6969537 0.7084025

(0.0567) (0.4681) (0.4224) (0.3867) (0.1088)

2 8 1.6319252 1.2117092 0.7764776 0.7758653 0.718633

(0.0992) (0.0901) (0.555) (0.4717) (0.0826)

3 8 1.5952712 1.3867556 0.8212589 0.8230702 0.7302381

(0.0932) (0.1393) (0.5945) (0.4568) (0.1935)

4 8 1.573993 1.3601346 0.8346759 0.7682195 0.8335882

(0.1033) (0.2176) (0.4628) (0.4301) (0.038)

5 8 1.7099923 1.3345994 0.826007 0.7964027 0.7935639

(0.0753) (0.0849) (0.4908) (0.5647) (0.3256)

1 9 1.6259966 1.0380794 0.7145935 0.688937 0.7505045

(0.0475) (0.5604) (0.4282) (0.3629) (0.0877)

2 9 1.6504422 1.2492783 0.804965 0.7637004 0.7488072

(0.0853) (0.0278) (0.5546) (0.4295) (0.0929)

3 9 1.6342375 1.3715148 0.8487656 0.8312616 0.7465864

(0.0925) (0.1978) (0.5094) (0.4991) (0.2119)

4 9 1.6187274 1.4101191 0.9537172 0.7778448 0.8393009

(0.0871) (0.3393) (0.1229) (0.425) (0.0413)

5 9 1.7426157 1.461582 0.8926487 0.8439849 0.7673679

(0.0657) (0.1448) (0.09) (0.5806) (0.2629)

1 10 1.5491352 1.0325499 0.7146293 0.691431 0.7731506

(0.0456) (0.5388) (0.42) (0.3693) (0.0592)

2 10 1.6502905 1.228216 0.832613 0.7801035 0.7949067

(0.0635) (0.0391) (0.5194) (0.4751) (0.085)

3 10 1.6436548 1.4643579 0.830816 0.8552365 0.8095547

(0.0869) (0.3126) (0.5357) (0.4927) (0.1442)

4 10 1.753154 1.4407767 0.9188797 0.7712108 0.9058

(0.036) (0.4205) (0.1666) (0.4173) (0.0967)

5 10 1.7678161 1.3867061 0.9239108 0.8124478 0.8503855

(0.0537) (0.1457) (0.9304) (0.5963) (0.1578)

1 11 1.5565231 1.043585 0.7304528 0.680885 0.8077332

(0.0413) (0.8266) (0.4208) (0.3432) (0.0561)

2 11 1.6593492 1.1353453 0.8400311 0.8002724 0.8212896

(0.0795) (0.2544) (0.452) (0.5012) (0.0671)

3 11 1.643109 1.4795633 0.9218304 0.9882717 0.8105468

(0.0953) (0.295) (0.1399) (0.536) (0.1444)

4 11 1.7557473 1.4062244 0.866866 0.7629437 0.9298047

(0.0259) (0.3579) (0.281) (0.3809) (0.1215)

5 11 1.8142915 1.2379518 0.8991062 0.8506304 0.9818662

(0.0257) (0.3242) (0.93) (0.6922) (0.0362)

1 12 1.5480611 1.0474744 0.7438101 0.6822789 0.8209919

(0.0676) (0.9277) (0.4219) (0.3605) (0.0632)

2 12 1.6944992 1.1588999 0.8963043 0.8039896 0.7708797

(0.0727) (0.2175) (0.2045) (0.5002) (0.0755)

3 12 1.6741854 1.3099761 0.9537163 0.9488489 0.7738955

(0.0746) (0.1927) (0.1219) (0.6575) (0.1548)

4 12 1.7010796 1.3645506 0.8935598 0.7620995 1.0608415

(0.0371) (0.3253) (0.1868) (0.4498) (0.2301)

5 12 1.8527262 1.2640211 1.1389237 0.9625845 1.1424182

(0.0386) (0.0682) (0.4515) (0.085) (0.0299)

Appendix D IPI Growth Forecast Using Fixed Window

Table 1

AO Random-Walk Model Number of quarters ahead

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

RMSE 0.111915 0.122462 0.122077 0.114456 0.135254

Table 2

Direct Autoregressive Model Number of periods ahead

Number of

lags

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

1 2.318737 1.250783 0.915867 0.851994 0.511659

(0.00075) (0.04791) (0.32597) (0.14298) (0)

2 2.372857 1.294724 0.978726 0.891125 0.517688

(0.00082) (0.04979) (0.46432) (0.23095) (0)

3 2.468914 1.245273 0.969661 0.895223 0.526134

(0.00113) (0.11866) (0.44129) (0.23417) (0.00001)

4 2.431552 1.246779 0.988979 0.903573 0.531885

(0.00634) (0.10185) (0.47863) (0.25499) (0.00003)

5 2.492654 1.212785 0.976159 0.916037 0.527053

(0.00674) (0.12589) (0.45376) (0.27948) (0.00009)

BIC 2.343673 1.287583 0.979833 0.867206 0.52839

(0.00062) (0.05114) (0.46236) (0.17432) (0)

Table 3

Factor Model Version (i) Number of periods ahead

Number of

lags

Number of

Factors

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

0 Bai-Ng

Test 2.302779 1.215523 0.995767 0.907495 0.529224

(0.01444) (0.05966) (0.49083) (0.2726) (0)

Table 4

Factor Model Version (ii) Number of periods ahead

Number of

lags

Number of

Factors

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

BIC Bai-Ng

Test 2.424432 1.243039 1.033486 0.963581 0.539125

(0.01727) (0.12766) (0.4409) (0.41265) (0)

Table 5

Factor Model Version (iii) Number of periods ahead

No Lags Number of

Factors

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

1 2.287181 1.272762 0.928425 0.842712 0.517968

(0.00404) (0.05994) (0.33085) (0.13131) (0)

2 2.325984 1.194515 0.974214 0.933081 0.536016

(0.01232) (0.09811) (0.44615) (0.33792) (0)

3 2.387296 1.172138 0.949354 0.939366 0.541492

(0.0114) (0.12538) (0.40368) (0.36009) (0)

4 2.39316 1.185536 0.945285 0.940848 0.530139

(0.00848) (0.10111) (0.3931) (0.37165) (0)

5 2.409928 1.184391 0.98373 0.967172 0.539584

(0.01012) (0.12566) (0.47151) (0.43148) (0)

6 2.478329 1.287979 1.10941 0.962375 0.549796

(0.00781) (0.08357) (0.35824) (0.41876) (0)

7 2.507593 1.320938 1.121481 0.950383 0.548514

(0.00622) (0.06193) (0.34138) (0.39189) (0)

8 2.565134 1.287052 1.154317 0.953755 0.553481

(0.00507) (0.0712) (0.30471) (0.39612) (0)

9 2.579179 1.29242 1.163307 0.997267 0.568814

(0.00463) (0.08112) (0.29723) (0.4934) (0)

10 2.513438 1.275846 1.160537 0.986032 0.583943

(0.00594) (0.08863) (0.30234) (0.46539) (0)

11 2.47019 1.229356 1.161593 0.976007 0.584522

(0.00545) (0.12324) (0.29134) (0.43715) (0)

12 2.392946 1.197087 1.148253 0.987234 0.596156

(0.00749) (0.17254) (0.30818) (0.46471) (0)

Table 6

Factor Model Version (iv) Number of periods ahead

Number of

lags

Number of

Factors

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

1 1 2.478061 1.223505 0.923842 0.857433 0.538753

(0.00102) (0.03804) (0.33535) (0.15654) (0)

2 1 2.301786 1.340806 1.055756 0.924388 0.518524

(0.00214) (0.04852) (0.40482) (0.30641) (0)

3 1 2.35415 1.294461 1.045017 0.927762 0.525642

(0.0052) (0.11423) (0.41053) (0.30453) (0.00001)

4 1 2.276635 1.294235 1.067356 0.939 0.527259

(0.01819) (0.08966) (0.36319) (0.32647) (0.00002)

5 1 2.310526 1.272 1.055497 0.947115 0.534071

(0.01633) (0.09461) (0.38832) (0.34534) (0.00011)

1 2 2.621768 1.107334 0.883275 0.834041 0.514245

(0.00162) (0.23493) (0.23218) (0.13155) (0)

2 2 2.286481 1.272515 1.047898 0.851512 0.518397

(0.00121) (0.05727) (0.41307) (0.13716) (0)

3 2 2.336144 1.242893 1.009306 0.862111 0.519643

(0.01638) (0.14423) (0.47917) (0.13572) (0)

4 2 2.351261 1.134828 1.019021 0.841992 0.533429

(0.02512) (0.22238) (0.45938) (0.1149) (0.00001)

1 3 2.605994 1.077391 0.850445 0.806686 0.526945

(0.00129) (0.30563) (0.20171) (0.10637) (0)

2 3 2.329769 1.298109 1.036471 0.838384 0.518192

(0.00092) (0.02961) (0.43163) (0.12972) (0)

3 3 2.348262 1.215425 0.976505 0.85764 0.53488

(0.01329) (0.17959) (0.45546) (0.15498) (0)

4 3 2.347628 1.12694 0.980631 0.835392 0.528396

(0.02292) (0.23886) (0.46582) (0.13287) (0.00001)

5 3 2.369906 1.416224 1.121372 1.004458 0.528144

(0.00314) (0.04481) (0.31162) (0.49059) (0.00001)

1 4 2.786979 1.062777 0.853823 0.826876 0.527468

(0.0056) (0.34499) (0.20287) (0.12333) (0)

2 4 2.345071 1.289454 1.046257 0.893557 0.521153

(0.00097) (0.02554) (0.41358) (0.1691) (0)

3 4 2.407148 1.172457 0.978769 0.957884 0.54912

(0.01394) (0.22269) (0.45931) (0.3379) (0)

4 4 2.404322 1.118394 1.010417 0.940063 0.546658

(0.01957) (0.26024) (0.48215) (0.2733) (0)

5 4 2.499153 1.484022 1.169309 1.090768 0.537657

(0.00168) (0.07227) (0.26198) (0.27524) (0.00003)

1 5 2.83388 1.15685 0.920973 0.827054 0.542398

(0.00146) (0.16586) (0.32257) (0.11906) (0)

2 5 2.433406 1.358623 1.163381 0.967385 0.529797

(0.00095) (0.01114) (0.24815) (0.36486) (0)

3 5 2.414455 1.203936 1.045435 1.004355 0.549126

(0.01582) (0.20961) (0.42845) (0.48379) (0)

4 5 2.431403 1.168887 1.120133 1.013361 0.552695

(0.01967) (0.20394) (0.3235) (0.45111) (0.00001)

5 5 2.469536 1.485984 1.205492 1.138157 0.540677

(0.00205) (0.07171) (0.18995) (0.20906) (0.00004)

1 6 2.833099 1.142383 0.888015 0.826422 0.552408

(0.00058) (0.17027) (0.23867) (0.13558) (0)

2 6 2.433338 1.472256 1.245753 1.052597 0.540934

(0.00162) (0.0182) (0.2236) (0.36227) (0)

3 6 2.382037 1.222231 0.969702 0.981753 0.556535

(0.02233) (0.19594) (0.43827) (0.43173) (0)

4 6 2.407493 1.276729 1.221349 1.091277 0.549313

(0.02583) (0.11726) (0.24288) (0.28821) (0.00001)

5 6 2.290308 1.529576 1.352706 1.240056 0.541292

(0.00264) (0.04851) (0.11912) (0.15808) (0.00008)

1 7 2.810144 1.182384 0.980939 0.931486 0.552078

(0.00054) (0.17494) (0.45365) (0.31302) (0)

2 7 2.398785 1.544477 1.26958 1.03604 0.558626

(0.00184) (0.01053) (0.21118) (0.41024) (0)

3 7 2.358812 1.243257 0.94685 1.042888 0.585946

(0.02416) (0.15577) (0.4036) (0.33162) (0)

4 7 2.42457 1.271416 1.235693 1.126368 0.574982

(0.02328) (0.10861) (0.24025) (0.22137) (0.00001)

5 7 2.330913 1.52611 1.32495 1.247734 0.589763

(0.002) (0.05017) (0.15295) (0.14309) (0.00083)

1 8 2.788621 1.210109 0.953807 0.914599 0.529047

(0.0005) (0.14929) (0.37742) (0.28962) (0)

2 8 2.474605 1.532041 1.30222 1.072283 0.532019

(0.00215) (0.01793) (0.18782) (0.32385) (0.00001)

3 8 2.337216 1.220633 0.926836 1.039626 0.571872

(0.02571) (0.18238) (0.36854) (0.32712) (0)

4 8 2.456241 1.243432 1.265924 1.112036 0.603164

(0.02101) (0.12932) (0.22798) (0.26704) (0.00018)

5 8 2.36141 1.609896 1.389605 1.250634 0.585077

(0.00122) (0.04051) (0.10495) (0.14304) (0.00064)

1 9 2.647032 1.232546 0.942922 0.917451 0.532668

(0.0001) (0.12703) (0.34564) (0.31097) (0)

2 9 2.445711 1.538123 1.304356 1.079229 0.53738

(0.0023) (0.01719) (0.18453) (0.28833) (0.00002)

3 9 2.341157 1.238356 0.941302 1.061719 0.586321

(0.02961) (0.1666) (0.38634) (0.21644) (0.00001)

4 9 2.440484 1.259386 1.246711 1.107949 0.596186

(0.02259) (0.11639) (0.22665) (0.26056) (0.00011)

5 9 2.325231 1.619675 1.40009 1.225651 0.588048

(0.00145) (0.02993) (0.09774) (0.13922) (0.00124)

1 10 2.648386 1.246474 0.936903 0.943792 0.55825

(0.00009) (0.1127) (0.32919) (0.36246) (0)

2 10 2.469031 1.53897 1.339495 1.075452 0.558784

(0.00177) (0.02274) (0.16151) (0.30758) (0)

3 10 2.331697 1.222462 0.947684 1.086824 0.590097

(0.02902) (0.1827) (0.40518) (0.18727) (0.00005)

4 10 2.423908 1.310085 1.225485 1.150883 0.598126

(0.01951) (0.0789) (0.25067) (0.19539) (0.00002)

5 10 2.275641 1.645929 1.46983 1.257995 0.589966

(0.00098) (0.02904) (0.06688) (0.09715) (0.00226)

1 11 2.636645 1.278185 0.972214 0.940008 0.542394

(0.00008) (0.06743) (0.42041) (0.37366) (0)

2 11 2.500471 1.500899 1.328236 1.073707 0.554438

(0.0016) (0.03391) (0.17037) (0.31529) (0)

3 11 2.363829 1.224427 0.947619 1.087786 0.592821

(0.02248) (0.18204) (0.40667) (0.1697) (0.00007)

4 11 2.352904 1.288209 1.230561 1.157903 0.650998

(0.0157) (0.11549) (0.24028) (0.1909) (0)

5 11 2.260823 1.649149 1.451874 1.251602 0.619856

(0.00096) (0.04132) (0.06896) (0.06243) (0.00325)

1 12 2.61764 1.26897 0.958413 0.901644 0.551818

(0.00002) (0.07314) (0.36418) (0.3161) (0)

2 12 2.556828 1.537699 1.319531 1.018978 0.562136

(0.00185) (0.01518) (0.16168) (0.45736) (0)

3 12 2.332926 1.228231 0.939502 1.110238 0.61686

(0.02381) (0.19643) (0.3949) (0.11741) (0.00001)

4 12 2.330109 1.271854 1.215345 1.105987 0.670404

(0.01102) (0.12825) (0.25788) (0.3119) (0)

5 12 2.279401 1.638254 1.454004 1.219 0.627205

(0.0007) (0.03996) (0.0555) (0.11955) (0.00609)

Appendix E IPI Growth Forecast Using Rolling Window

Table 1

AO Random-Walk Model Number of quarters ahead

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

RMSE 0.111915 0.122462 0.122077 0.114456 0.135254

Table 2

Direct Autoregressive Model Number of periods ahead

Number of

lags

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

1

2.30422 1.213299 0.897886 0.841357 0.523612

(0.00098) (0.21739) (0.42498) (0.42287) (0.06327)

2

2.398289 1.289219 0.978382 0.91639 0.54173

(0.0009) (0.15725) (0.28062) (0.47615) (0.01739)

3

2.601302 1.274025 1.053907 0.95177 0.569741

(0.00235) (0.01996) (0.40844) (0.94362) (0.16122)

4

2.622283 1.339035 1.078532 0.978107 0.582727

(0.00018) (0.03664) (0.23102) (0.68791) (0.39225)

5

2.847081 1.409442 1.130601 1.073264 0.618165

(0.11346) (0.41151) (0.62008) (0.48687) (0.16876)

BIC

2.353 1.341341 1.088674 0.986252 0.621465

(0.07896) (0.18029) (0.65939) (0.37967) (0.12454)

Table 3

Factor Model Version (i) Number of periods ahead

Number of

lags

Number of

Factors

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

0 Bai-Ng

Test 2.377937 1.2967 0.851299 0.773992 0.544998

(0.07896) (0.18029) (0.65939) (0.37967) (0.12454)

Table 4

Factor Model Version (ii) Number of periods ahead

Number of

lags

Number of

Factors

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

BIC Bai-Ng

Test 3.440653 1.891393 1.492572 1.023374 0.640193

(0.01287) (0.01253) (0.13583) (0.1726) (0.06106)

Table 5

Factor Model Version (iii) Number of periods ahead

No Lags Number of

Factors

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

1 2.25671 1.226911 0.916234 0.850441 0.563657

(0.04603) (0.04871) (0.92269) (0.81022) (0.13156)

2 2.402692 1.344415 0.966312 0.879066 0.522694

(0.07082) (0.08143) (0.89746) (0.36132) (0.13033)

3 2.427535 1.397815 0.993428 0.885498 0.525647

(0.06687) (0.0991) (0.77099) (0.50652) (0.12836)

4 2.561374 1.433352 0.914335 0.82002 0.516027

(0.11943) (0.50555) (0.87385) (0.3335) (0.09659)

5 2.54497 1.53438 0.916907 0.812243 0.511202

(0.07854) (0.48528) (0.70696) (0.49997) (0.08239)

6 2.60909 1.550732 0.95316 0.819891 0.49813

(0.05217) (0.42099) (0.65434) (0.44564) (0.05661)

7 2.623027 1.558043 0.991293 0.85061 0.520029

(0.05106) (0.41747) (0.71209) (0.47009) (0.06541)

8 2.509469 1.585167 1.007555 0.853542 0.512395

(0.06474) (0.35484) (0.81764) (0.5531) (0.06935)

9 2.519252 1.613203 1.004842 0.871251 0.519044

(0.04884) (0.33489) (0.79277) (0.42742) (0.12359)

10 2.563141 1.597731 1.00028 0.87415 0.524453

(0.04894) (0.33089) (0.77925) (0.81577) (0.12054)

11 2.678076 1.659914 0.985833 0.873141 0.524988

(0.05701) (0.22812) (0.93322) (0.94697) (0.12346)

12 2.705526 1.715351 0.958331 0.86296 0.514536

(0.05143) (0.19569) (0.87694) (0.91799) (0.11862)

Table 6

Factor Model Version (iv) Number of periods ahead

Number of

lags

Number of

Factors

1 quarter 2 quarter 3 quarter 4 quarter 8 quarter

1 1 3.398468 1.767145 1.285923 0.969059 0.56387

(0.01165) (0.00572) (0.26597) (0.15348) (0.05029)

2 1 3.377153 1.894319 1.58778 1.139547 0.603002

(0.01075) (0.18872) (0.0206) (0.50998) (0.0198)

3 1 3.322196 1.77861 1.827876 1.347516 0.622243

(0.0137) (0.06615) (0.10405) (0.01592) (0.0172)

4 1 3.368682 1.770439 1.836798 1.388301 0.620682

(0.01735) (0.08197) (0.10517) (0.00092) (0.02299)

5 1 3.508041 1.895216 1.874205 1.361988 0.633173

(0.01644) (0.18189) (0.16033) (0.00382) (0.00073)

1 2 3.417657 1.740823 1.187546 0.912986 0.588337

(0.01298) (0.02013) (0.31233) (0.84341) (0.0597)

2 2 3.299513 1.901604 1.5666 1.040551 0.658555

(0.00858) (0.20439) (0.06557) (0.6681) (0.03481)

3 2 3.304853 1.764427 1.784419 1.305226 0.626005

(0.01553) (0.06484) (0.07464) (0.0405) (0.01586)

4 2 3.350515 1.715954 1.760072 1.321891 0.660471

(0.01575) (0.06692) (0.14678) (0.00563) (0.01356)

1 3 3.415567 1.687621 1.173445 0.922708 0.588768

(0.01279) (0.00624) (0.20054) (0.78816) (0.05653)

2 3 3.327388 1.86556 1.488291 1.026808 0.666754

(0.00905) (0.16913) (0.01593) (0.694) (0.04036)

3 3 3.335011 1.730117 1.73623 1.306436 0.617303

(0.01682) (0.05557) (0.03177) (0.04116) (0.01232)

4 3 3.333733 1.69643 1.716658 1.331709 0.66959

(0.01533) (0.0709) (0.09302) (0.00608) (0.01771)

5 3 3.422678 1.831183 1.718526 1.292785 0.6307

(0.02047) (0.22727) (0.12758) (0.00085) (0.00049)

1 4 3.330663 1.69052 1.133567 0.861728 0.588054

(0.01202) (0.10613) (0.18427) (0.63472) (0.09692)

2 4 3.241797 1.859774 1.462272 1.051177 0.680919

(0.0092) (0.1879) (0.0325) (0.64419) (0.0466)

3 4 3.353818 1.697369 1.723995 1.316225 0.608628

(0.0178) (0.07158) (0.02008) (0.05932) (0.01733)

4 4 3.336173 1.671332 1.687111 1.349955 0.64611

(0.01628) (0.07674) (0.08179) (0.00037) (0.01737)

5 4 3.471196 1.785406 1.673688 1.263932 0.631525

(0.02898) (0.24246) (0.04131) (0.001) (0.00035)

1 5 3.377314 1.747319 1.140624 0.833806 0.582478

(0.01448) (0.08452) (0.26463) (0.69649) (0.0914)

2 5 3.275017 1.881975 1.487444 1.056989 0.67582

(0.01177) (0.18354) (0.02918) (0.64144) (0.04519)

3 5 3.453306 1.705846 1.710288 1.278094 0.624109

(0.02494) (0.08037) (0.01966) (0.05279) (0.01737)

4 5 3.356484 1.645424 1.681429 1.377558 0.639372

(0.01811) (0.06955) (0.09979) (0.0007) (0.02238)

5 5 3.408254 1.778833 1.621173 1.260435 0.664444

(0.02601) (0.2501) (0.02751) (0.00197) (0.01711)

1 6 3.360553 1.787002 1.145664 0.826015 0.591188

(0.01329) (0.06973) (0.18391) (0.67778) (0.1164)

2 6 3.295756 1.887435 1.523579 1.032944 0.671909

(0.0115) (0.1933) (0.01825) (0.55158) (0.07199)

3 6 3.444527 1.669846 1.759532 1.343427 0.630684

(0.02407) (0.09497) (0.02027) (0.06466) (0.0188)

4 6 3.35236 1.633397 1.791659 1.440072 0.650909

(0.01682) (0.07853) (0.09177) (0.00005) (0.04698)

5 6 3.381405 1.755551 1.704264 1.355118 0.676776

(0.02897) (0.24235) (0.02776) (0.00158) (0.0544)

1 7 3.327608 1.772593 1.13697 0.851707 0.600592

(0.01438) (0.09464) (0.17099) (0.54424) (0.11877)

2 7 3.296199 1.896844 1.519608 1.011571 0.668279

(0.01053) (0.20379) (0.02839) (0.55626) (0.06585)

3 7 3.421647 1.693513 1.750376 1.366509 0.684406

(0.0247) (0.10687) (0.01819) (0.0315) (0.00316)

4 7 3.32183 1.626138 1.799707 1.443522 0.710009

(0.01856) (0.07317) (0.06924) (0.0005) (0.01401)

5 7 3.349533 1.718345 1.760989 1.291031 0.686701

(0.02854) (0.20778) (0.08018) (0.00052) (0.03491)

1 8 3.351487 1.789504 1.155755 0.855412 0.623446

(0.01395) (0.111) (0.4391) (0.47366) (0.12376)

2 8 3.297842 1.900665 1.52663 1.046275 0.728309

(0.00955) (0.20101) (0.03675) (0.48497) (0.04359)

3 8 3.455872 1.695137 1.745275 1.364723 0.708788

(0.02466) (0.09954) (0.01647) (0.08307) (0.00507)

4 8 3.338161 1.609847 1.758501 1.400536 0.698676

(0.02475) (0.08781) (0.05629) (0.00189) (0.01473)

5 8 3.342007 1.788708 1.746579 1.282177 0.703475

(0.03398) (0.20953) (0.02987) (0.00096) (0.01498)

1 9 3.374867 1.773157 1.173255 0.834477 0.636764

(0.01298) (0.09411) (0.42018) (0.38021) (0.14797)

2 9 3.326817 1.894498 1.506731 1.044915 0.724331

(0.00901) (0.18514) (0.08369) (0.47639) (0.04417)

3 9 3.446655 1.688704 1.705608 1.357751 0.688655

(0.02619) (0.03905) (0.06657) (0.06566) (0.00656)

4 9 3.334194 1.606997 1.712715 1.385993 0.712224

(0.02455) (0.06899) (0.05194) (0.01513) (0.01789)

5 9 3.343078 1.760246 1.625537 1.278929 0.780544

(0.03633) (0.17253) (0.0308) (0.00036) (0.02647)

1 10 3.353857 1.678615 1.160262 0.829761 0.647014

(0.01262) (0.09641) (0.51237) (0.36162) (0.14898)

2 10 3.326697 1.901321 1.522625 1.014286 0.740106

(0.0091) (0.18607) (0.08443) (0.56657) (0.03989)

3 10 3.464704 1.718208 1.67932 1.389566 0.691157

(0.02521) (0.04615) (0.06636) (0.01244) (0.00417)

4 10 3.317848 1.628034 1.696254 1.359786 0.706187

(0.02532) (0.08181) (0.0445) (0.00663) (0.02357)

5 10 3.332849 1.747871 1.634155 1.283324 0.7186

(0.03934) (0.19684) (0.02021) (0.00004) (0.03277)

1 11 3.364461 1.730072 1.178409 0.826434 0.656508

(0.01333) (0.05548) (0.35556) (0.26726) (0.14154)

2 11 3.296029 1.951656 1.564787 1.097152 0.734515

(0.00939) (0.20082) (0.10787) (0.0462) (0.03285)

3 11 3.458068 1.780519 1.628312 1.398343 0.693872

(0.02602) (0.04098) (0.05171) (0.00438) (0.0077)

4 11 3.287124 1.568681 1.70696 1.359834 0.722983

(0.0253) (0.08401) (0.0435) (0.0088) (0.02276)

5 11 3.252039 1.703218 1.708251 1.373014 0.826061

(0.03424) (0.16888) (0.01671) (0.01162) (0.13632)

1 12 3.366698 1.816795 1.223046 0.835183 0.654286

(0.01323) (0.0389) (0.45309) (0.23639) (0.13644)

2 12 3.238194 2.028269 1.498884 1.126216 0.737504

(0.00928) (0.16359) (0.08153) (0.01097) (0.05389)

3 12 3.421456 1.693079 1.572251 1.365439 0.698172

(0.02419) (0.08087) (0.02463) (0.00883) (0.00719)

4 12 3.279827 1.549486 1.683885 1.334704 0.781332

(0.02248) (0.10798) (0.05664) (0.02843) (0.03032)

5 12 3.184654 1.793619 1.718191 1.500431 0.88414

(0.03052) (0.12904) (0.01238) (0.00036) (0.40221)