# Chapter 12: Time Series Models

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2 Testing for serial correlation in Koyck distributed lag models using Durbin s h test (UE ): Estimate the Koyck distributed lag model before attempting this section (i.e., Equation EQ02 should already be present in the workfile). To conduct a Durbin s h test for UE, Equation 12.11, follow these steps: Step 2. To determine whether the value in parenthesis, in the denominator under the square root sign in UE, Equation 12.17, is positive, enter the following command in the command window: scalar Press Enter to create a scalar object named denominator. Double click the scalar object icon named denominator in the EViews workfile and view its value in the left corner of the status bar (bottom of the EViews window). If the number is positive, continue with the next step; if not, Durbin s h test is not valid. Step 3. To compute Durbin s h test statistic shown in UE, Equation 12.17, enter the following command in the command window and press Enter: scalar Step 4. To view this scalar, double click the scalar object icon named dhtest and view its value in the left corner of the status bar (bottom of the EViews window). If the number is 1.96, reject the null hypothesis of no first order serial correlation. Testing for serial correlation in Koyck distributed lag models using the Lagrangian Multiplier (LM) (UE ): Estimate the Koyck distributed lag model before attempting this section (i.e., equation EQ02 should already be present in the workfile). To conduct a Lagrangian Multiplier (LM) test for UE, Equation 12.11, follow these steps: Step 1. Open the EViews workfile named Macro14.wf1. Step 2. Open the Equation named EQ02 by double clicking its icon in the workfile window. Step 3. Select View/Residuals Tests/Serial Correlation LM Test on the equation menu bar (see highlighted selections in the graphic on the right).

3 Step 4. Change the number in the Lags to include: to 1 in the Lag Specification: window. 2 Click OK to reveal the following EViews output: Breusch-Godfrey Serial Correlation LM Test: F-statistic Probability Obs*R-squared Probability Test Equation: Dependent Variable: RESID Method: Least Squares Date: 07/01/00 Time: 10:25 Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-statistic Prob. C YD CO(-1) RESID(-1) R-squared Mean dependent var 3.67E-14 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) The null hypothesis of the LM test is that there is no serial correlation up to lag order p, where p is equal to 1 in this case. The Obs*R-squared statistic is the Breusch-Godfrey LM test statistic. This LM statistic is computed as the number of observations times the R 2 from the test regression. The LM test statistic is asymptotically distributed as a χ 2 with p degrees of freedom (p is equal to 1 in this case). Step 5. To determine whether the null hypothesis can be rejected in this case, determine the critical χ 2 (1) value from UE Table B-8. The critical χ 2 value can also be calculated in EViews by typing the following formula in the EViews command window: 3 EViews returns a scalar value of Since the calculated Breusch-Godfrey LM test statistic of 9.42 exceeds the critical χ 2 (1) value, we can reject the hypothesis of no serial correlation up to lag order 1 at the 95% confidence level. The probability printed to the right of the Obs*R-squared statistic in the EViews output (i.e., ) represents the probability that you would be incorrect if you rejected the null hypothesis of no serial correlation up to lag order 1 at the 95% confidence level. 2 EViews enters 2 lags by default (i.e., testing for second order serial correlation). We will enter 1 lag to estimate the LM statistic for UE, Equation 12.20, p calculates the percentile of the χ 2 distribution. The formula finds the value c such that the prob(χ 2 with v degrees of freedom is c) = p. In this case, the prob(χ 2 with 1 degree of freedom is 3.84) =95%. In other words, 95% of the area of a χ 2 distribution, with v=1 degrees of freedom, is in the range from 0 to 3.84 and 5% is in the range from 3.84 to (i.e., in the tail of the distribution).

4 Performing Granger Causality tests (UE 12.3): To conduct Granger Causality test for CO and YD, follow these steps: Step 2. To Create an EViews group for the current purchases of goods and services (CO) and the level of disposable income (YD), hold down the Ctrl button, click on, CO and YD, select Show from the workfile toolbar, and click OK. Step 3. Select Name on the Group Object menu bar, enter GROUP01 in the Name to identify object: window, and click OK. Step 4. Select View/Granger Causality on the Group Object menu bar. When you select the Granger Causality view, you will first see a dialog box asking for the number of lags to use in the test regressions. Change the number in the Lags to include: to 3 in the Lag Specification: window, and click OK. 4 EViews returns the following pairwise Granger Causality Tests Table. Pairwise Granger Causality Tests Date: 07/01/00 Time: 14:43 Sample: Lags: 3 Null Hypothesis: Obs F-Statistic Probability YD does not Granger Cause CO CO does not Granger Cause YD Step 5. Based on the Probability values reported in the table, the hypothesis that YD does not Granger Cause CO cannot be rejected, but the hypothesis that CO does not Granger cause YD can be rejected. Therefore, it appears that Granger causality runs one way, from CO to YD, but not the other way. 5 Testing for nonstationarity by calculating the auto correlation function ACF (UE , Equation 12.24, p. 425): Follow these steps to calculate the auto correlation function ACF: Step 2. Open CO in one window by double clicking the series icon in the workfile window. Step 3. To view the Autocorrelation and Partial Correlation, Select View/Correlogram, on the CO series menu bar and a Correlogram Specification dialog box appears. Select level in the Correlogram of: window and enter 16 (the EViews default in this case) in the Lag Specification: lags to include: window, and click OK to reveal the EViews output below. 4 In general, it is better to use more rather than fewer lags, since the theory is couched in terms of the relevance of all past information. You should pick a lag length that corresponds to reasonable beliefs about the longest time over which one of the variables could help predict the other. 5 The reported F-statistics are the Wald statistics for the joint hypothesis that the coefficients on the lagged values of the other variable are zero for each equation (CO in the first equation and YD in the second equation for the table printed above). In case you want to determine significance by comparing the calculated F statistic with the critical F value from the F Table, the numerator degrees of freedom are given by the number of coefficient restrictions in the null hypothesis (i.e., the number of lags) and the denominator degrees of freedom are given by the total regression degrees of freedom.

5 Step 4. Since the AC's are significantly positive and the AC(k) dies off geometrically with increasing lag k, it is a sign that the series obeys a low-order autoregressive (AR) process. 6 In addition, since the partial autocorrelation (PAC) is significantly positive at lag 1 and close to zero thereafter, the pattern of autocorrelation can be captured by an autoregression of order one (i.e., AR(1)). The finding in Steps 3 & 4 indicates that the CO series violates the third criteria for stationarity (UE, top of p. 425) and provides strong evidence that the CO series is non-stationary. Testing for nonstationarity with the Dickey-Fuller (DF) test (12.4.2): Since the AC analysis in the previous section indicated that CO is most likely an AR(1) process, the Dickey-Fuller (DF) test is valid. Follow these steps to conduct the Dickey-Fuller test of the hypothesis that the CO series is non-stationary: Step 2. Open CO in one window by double clicking the series icon in the workfile window. Note that EViews will probably display the correlogram view for CO since that was the last view selected in the previous section. You can select View/Spreadsheet to view the series data or just proceed with the next step to test for stationarity. Step 3. To conduct the Dickey-Fuller (DF) test, select View/Unit Root Test on the CO series window menu bar. Step 4. Four things have to be specified in the Unit Root Test dialog box to carry out a unit root test. First, choose the type of test either the Augmented Dickey-Fuller (ADF) test or the Phillips- 6 If the AC(1) is nonzero, it means that the series is first order serially correlated. If AC(k) dies off more or less geometrically with increasing lag k, it is a sign that the series obeys a low-order autoregressive (AR) process. If AC(k) drops to zero after a small number of lags, it is a sign that the series obeys a low-order moving-average (MA) process. EViews also reports the Partial Correlations (PAC) in the same window. The partial correlation at lag k measures the correlation of CO values that are k periods apart, after removing the correlation from the intervening lags. If the pattern of autocorrelation is one that can be captured by an autoregression of order less than k, then the partial autocorrelation at lag k will be close to zero. The PAC of a pure autoregressive process of order k cuts off at lag k, while the PAC of a pure moving average (MA) process asymptotes gradually to zero.

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