CHAPTER 16

Damodar GujaratiEconometrics by Example, second editionCHAPTER 16ECONOMIC FORECASTING

Damodar Gujarati

Econometrics by Example

TIME SERIES ECONOMETRICSFour important topics in time series econometrics are:(1) Forecasting with linear regression models(2) Univariate time series forecasting with Box-Jenkins methodology (ARIMA)(3) Multivariate time series forecasting using vector autoregression(4) The nature of causality in econometrics

Damodar GujaratiEconometrics by Example, second edition

Damodar Gujarati


FORECASTING WITH LINEAR REGRESSION MODELSLong been used in forecasting sales, production, employment, corporate profits and host of other economic topics.Point and interval forecasts: In point forecast a single value for each forecast period is provided, whereas in interval forecast we obtain a range, or an interval, that will include the realized value with some probability.Ex-post and ex-ante forecasts: In the ex-post forecast period (the holdover period), we also know the values of the regressand and regressors.Conditional and unconditional forecasts: In conditional forecasts (scenario analysis or contingency analysis), we forecast the variable of interest conditional on the assumed values of the regressors. In unconditional forecasts, we know the values of the regressors with certainty instead of picking some arbitrary values of them, as in conditional forecasting.


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ARIMA METHOD OF FORECASTINGThe Autoregressive (AR) ModelThe following is an AR(p) model: where ut is a white noise error term.

The Moving Average (MA) ModelWe can also model Yt as an the MA(q) model, a weighted, or moving, average of the current and past white noise error terms:


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ARIMA METHOD OF FORECASTING (CONT.)The Autoregressive Moving Average (ARMA) ModelThe ARMA (p,q) model is a combination of AR (autoregressive) and MA (moving average) terms.

The Autoregressive Integrated Moving Average (ARIMA) ModelThe BJ methodology is based on the assumption that the underlying time series is stationary or can be made stationary by differencing it one or more times. This is known as the ARIMA (p, d, q) model, where d denotes the number of times a time series has to be differenced to make it stationary.


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ARIMA METHOD OF FORECASTING (CONT.)The BJ methodology to determine which model is appropriate follows a four-step procedure:Step 1: Identification: Determine the appropriate values of p, d. and q.The main tools in this search are the correlogram and partial correlogram.Step 2: Estimation: Estimate the parameters of the chosen model. Step 3: Diagnostic Checking: Check if the residuals from the fitted model are white noise.If they are, accept the chosen model; if not, start afresh.That is why the BJ methodology is an iterative process. Step 4: Forecasting. The ultimate test of a successful ARIMA model lies in its forecasting performance, within the sample period as well as outside the sample period.


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VECTOR AUTOREGRESSION (VAR)Vector autoregressive models (VARs) are used to deal with forecasting two or more time series.In VAR we have one equation for each variable and each equation contains only the lagged values of that variable and the lagged values of all other variables in the system. As in the case of the univariate time series, in VAR we also require the time series to be stationary.If each variable in the VAR is already stationary, each equation in it can be estimated by OLS. If each variable is not stationary, we can estimate VAR only in the first-differences of the series.If individual variables in VAR are nonstationary, but are cointegrated, we can estimate VAR by taking into account the error correction term, which is obtained from the cointegrating regression. This leads to vector error correction model (VECM).


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NATURE OF CAUSALITYVAR modes can also be used to shed light on causality among variables. The basic idea behind VAR causality testing is that the past can cause the present and the future, but not the other way around. Granger causality: In establishing causality, we must make sure that the underlying variables are stationary. If they are not, we have to difference the variables and run the causality test on the differenced variables. However, if the variables are nonstationary, but are integrated, we need to use the error correction term to account for causality, if any.


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GRANGER CAUSALITY TEST1. Regress current Y on all lagged Y terms and other variables, if any (such as trend), but do not included the lagged X terms in this regression. This is the restricted regression. Obtain the restricted residual sum of squares, RSSr.2. Reestimate the equation including the lagged X terms. This is the unrestricted regression. From this regression obtain the unrestricted residual sum of squares, RSSur.3. The null hypothesis is that the lagged X terms do not belong in the regression.4. To test the null hypothesis, we apply the F test, which is:

which has m and (n-k) df, where m is the number of lagged X terms, k is the number of parameters estimated in the unrestricted regression, and n is the sample size. 5. If the computed F value exceeds the critical F value at the chosen level of significance, we reject the null hypothesis.


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GRANGER CAUSALITY TEST (CONT.)Run the test again switching X and Y. There are four cases:1. Unidirectional causality from X to Y2. Unidirectional causality from Y to X3. Feedback or Bilateral causality is indicated when the sets of Y and X coefficients are statistically significantly different from zero in both regressions.4. Independence is suggested when the sets of Y and X coefficients are not statistically significant in either of the regressions.


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CHAPTER 16