Question

Suppose that the Gross Domestic Product (GDP) in the US is denoted as G_t. Let the quarterly data of G_t from 1980:1 to 2015:1 is non-stationary but the first difference of G_t, denoted as DG_t, is stationary. Assume that a researcher identified the following AR model for DG_t:

DG_t = Alpha₀ + Alpha₁DG_t-1 + Alpha₃DG_t-3 + Alpha₄DG_t-4 + e_t

Suppose the estimated results of the above AR model are as follows:

DG_t = 6.09 + 0.18DG_t-1 + 0.12DG_t-3 + 0.05DG_t-4

(a) Based on what criteria of the coefficients of auto-correlation function (ACF) and the coefficients of partial auto-correlation function (PACF), the researcher identified the above AR model? Explain. Also explain why the researcher chose 1, 3, and 4 lags.

(b) Forecast the GDP for 2015:2, assuming that the GDP in 2015:1, 2014:4, 2014:3, 2014:2, 2014:1, 2013:4, 2013:3, and 2013:2 respectively are: 16264.1, 16294.7, 16205.6, 16010.4, 15831.7, 15916.2,15779.9, and 15606.6.

(c) Explain how to conduct a diagnostic test to check if the researcher has identified the correct AR model. (Hint: Step 3 of Box-Jenkin’s Method)

Please show work.

Answer 1

1. The researcher identified the AR model with maximum order 4 as the ACF values tend to decline slightly post 4 and the PACF values sharply cut off post the value of 4. However, in the model they took only lags 1,3 and 4 as the lag 2 coefficient did not come significant.
2. Change in 2015Q1 value (lag 1)=16264.1-16294.7= -30.6

Change in 2014Q3 value (lag 3)=16205.6-16010.4=195.2

Change in 2014Q2 value (lag 4)= 16010.4-15831.7=178.7

Thus the predicted change in 2015Q2 value is = 6.09 + 0.18*(-30.6) + 0.12*195.2 + 0.05*178.7 =32.941

Thus predicted value in 2015Q2 =value in 2015 Q1 + predicted change in value in 2015Q2

=16264.1+32.941=16297.04

3. Portmanteau test or ljung box test is used to identify if the correct AR model is used.

The test statistic is given by :

sample size * (sum of squares of the various order auto correlations of the estimated residuals)

Here, the various orders are from 1 to K where K is chose sufficiently large.

Here under null hypothesis the test should follow a chi square distribution with K-4 degrees of freedom since here AR 4 model is used.

If the null hypothesis is rejected it means the residuals are correlated, implies the AR model is not correct.