The Art of Forecasting

Sam Ouliaris

NUS Business School &Monetary Authority of Singapore

Forecasting … Definition

Forecasting is the art of predicting the future value of a random variable (i.e., a variable with more than one possible outcome).

Amalgam of Numerous Disciplines

Forecasting uses tools from many disciplines (statistics, economics, computer science).

Forecasts Often Involve Subjective Judgment

Why? Professional judgment often improves on a forecast derived using formal techniques.

Aim of Forecasting In short, forecasting aims to predict

the future values of a random variable as accurately as possible.

We usually prepare these forecasts using all (or any part of) the relevant information available when the forecast is being prepared.

Forecast is Tough. So, Why Bother? Two reasons:

uncertainty about the future; the lags involved in decision making.

The environment we operate in is constantly changing. Forecasts help us by predicting these changes ahead of time.

Critical decisions can then be made based on rational expectations of future conditions.

Why Do We Forecast? In particular, forecasting often provides

decision makers with time to react to unwanted outcomes (e.g., an economic recession).

Forecasting facilitates the formulation of rational / consistent economic policies that help the Singapore economy achieve sustainable economic growth.

Lags in Decision Making… If we could always adjust instantaneously and

costlessly to new conditions there would be no need for forecasts.

But decisions rarely have immediate effects. In some cases, e.g., investment decisions, the changes take months, or years, to implement.

Organisations must therefore plan ahead, and to do this they need forecasts of future conditions.

Quantitative Information Needed For example, Singapore’s exchange rate

policy relies on a sequence of forecasts of future GDP. To this end, the MAS has developed a model of the Singapore economy that predicts the future value of GDP.

Qualitative feel is not good enough for policy formulation; policy makers need quantitative values to properly formulate exchange rate policy.

Forecasting: A Formal Definition Let the future (unknown value) of a

random variable, X, be denoted by Pt+n.

Note that “t” represents the present, and t+n denotes “n” periods ahead of “t”.

For example, if t = 2003, and n = 2, we aim to predict the value of X in year 2005 (= t+n).

Information? What’s That? Historical behaviour of the variable itself,

i.e., Xt, for t = 1 … n.

Historical behaviour of other variables that we think might affects the value of Xt. Economic theory plays a critical role here.

Intuition (subjective judgment), which typically starts with the historical behaviour of the variable.

Let’s Get Technical In general, a forecast may be represented

as Pt+n = f(Z1t,Z2t,a,b), where Z1t and Z2t are variables that potentially affect the future value, and “a” and “b” are parameters that reflect the strength of the relationship between the variable of interest, X, and Z1t and Z2t.

Typically, “a” and “b” need to be estimated for this model to be operational.

Evaluating Forecasts

We traditionally examine the size of the forecast errors, or Xt+j - Pt+j for j = 1,…,n.

This can be stated as “actual – predicted” for each of the periods we prepared the forecast for X.

The smaller these are, the better.

Don’t be Too Demanding

The public tends to evaluate a forecaster on how close the most recent forecast is to the correct value.

Not very reasonable! Why? The probability of predicting the actual value of a continuous random variable is ZERO!

A Good Forecaster Is… Someone who consistently generates small (not

necessarily the smallest) forecast errors over time. In other words, he/she is “reasonably” accurate on average.

Allowed to have bad calls --- they are sure to happen according to basic statistics.

Someone who provides a range of possible outcomes, and assigns a probability that the range will have the value he/she is trying to predict.

Other Important Criteria for Evaluating Forecasts

Forecasts need to be:

timely; cost effective; consistent; and, comprehensible by decision makers.

Forecasting Horizon Complicates Matters The further out you go in the forecast

horizon, the harder it is to predict accurately.

Uncertainty tends to compound itself, particularly as we extend the forecast horizon.

Three Approaches to Forecasting Econometrics, which is an amalgam of

economic theory (“econ”) and statistics (“metrics”).

Time-Series Analysis, which uses only the past behaviour of a variable to predict itself.

A combination of the previous two, plus subjective judgment.

Econometrics Starts with an economic

relationship that connects one variable to another. These statements imply causality.

For example, per capita consumption tends to move with disposable household income.

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Plot of Log(C) against Log(Y)

Keynesian Consumption Function

We can express this causal statement mathematically as

Log(C) = a + bLog(Y)+e,

where a > 0 and 0 < b < 1, are unknown parameters, and e is the “residual” that is meant to be unpredictable.

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Plot of Log(C) against Log(Y)

Role of Econometrics

It is the role of econometrics to estimate the unknown parameters, “a” and “b”, given the assumption that Log(C) = a + bLog(Y) + e.

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Keynesian Consumption Function for the USA

We collected actual time-series data on USA’s “real” consumption and household disposable income (1947:1 to 2002:04, quarterly) and obtained the following Keynesian consumption function:

C = -82.07178 + 0.92596Y,

Degree of Fit: 99.7 per cent.

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Forecasting with the USA Consumption Function

Household disposable income in 2003:01 was reported to be 7164.97.

If we substitute this value of income into the estimated consumption function to forecast consumption for 2003:01, we would obtain a predicted value of 6552.41. The actual value was 6713.58.

This implies a forecast error of 161.17 million (2.5% error relative to actual).

It’s certainly premature to assess the value of this model --- we presently have only one forecast error, and the probability that this error will take on any particular value is formally ZERO!

Caveats The consumption function we estimated is quite

simplistic.

Interest rates, wealth can also affect current consumption, and these variables in turn are driven by other variables.

In other words, predicting an economic variable often requires more than one equation. We need to formulate a model of the entire economy (i.e., a system of simultaneous equations).

Time-Series Analysis Time-Series Analysis uses the previous

behavior of a variable to predict itself.

Includes a number of well-known forecasting techniques:

simple smoothing, trend extrapolation, and decomposition models;

ARIMA (Box-Jenkins) models; and, other univariate and multivariate statistical methods (e.g.,

Vector Autoregression models), Artificial Neural Networks approaches.

Box-Jenkins (BJ) Example For a very simple BJ example,

consider Log(Ct) = a + bLog(Ct-1),

where “a” and “b” are unknown parameters.

Notice that we do not rely on any other variable to predict Ct.

Box-Jenkins Model for USA’s Consumption, 1947:2-2002:4 Log(Ct)=0.0090469+0.9999Log(Ct-1).

Very good fit of 99.9%.

Predicted C for 2003:01 is 6702.96 billion USD (1996, $); actual reported value is 6713.58.

Forecast error is quite small, 0.16%.

Subjective Judgment The final approach to forecasting includes

the use of subjective judgment in whole or in part.

This approach typically involves using the formal numbers derived from econometric estimation and/or time-series analysis, and then adjusting the prediction for qualitative information that is difficult to incorporate in a formal setting.

Skills Needed to Arrive Here…

Incidentally, to get to this point yourselves you need to master some:

economic theory; time-series data collection techniques; basic understanding of how to calculate a regression using standard

regression packages (computing); statistical theory; and, interpretation skills.

Most of these skills are acquired at university if you decide to major in economics (highly recommended --- or at least a minor!).