Question

Consider time series yt , defined as the daily percentage change in SP500 index. A researcher estimated the following model:

(a) There is one partial autocorrelation coefficient that you can find from the estimation result. What is the value of it? What is order (k ) of it?

Dependent Variable: GROWTH Method: Least Squares Date: 03/08/15 Time: 15:25 Sample (adjusted): 3/31/2011 6/08/2012 Included observations: 301 after adjustments Variable Coefficient Std. Error t-Statistic Prob. GROWTH(-1) GROWTH(-2) GROWTH(-3) 0.007810 -0.053530 0.102137 -0.152021 0.076356 0.057316 0.057393 0.057634 0.102283 0.9186 |-0.933949 0.3511 1.779602 0.0762 -2.637699 0.0088 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic) 0.041516 Mean dependent var 0.031834 S.D. dependent var 1.324685 Akaike info criterion 521.1724 Schwarz criterion -509.7206 Hannan-Quinn criter. 4.288123 Durbin-Watson stat 0.005549 0.007331 1.346287 3.413426 3.462690 3.433139 1.989886

(b) Test the null hypothesis that the partial autocorrelation coefficient that you have is zero against the alternative that it is not zero.

Answer 1

(a)

the regression has growth in period t as dependent variable and growth in period t-1,t-2 and t-3 as independent variables. The regression equation is as

$y_{t}=\beta _{0}+\beta {_{1}}y_{t-1}+\beta {_{2}}y_{t-2}+\beta {_{3}}y_{t-3}+u_{t}$

partial autocorrelation coefficients are the values estimated in this regression equation as given in the ANOVA table.

out of the three estimates, $\beta {_{3}}$ is only significant with negligible pvalue and tstatistics of 2.638 as it is greater than 2 implies the result is significant at 5%level of significance.

the coefficient value is -0.1520 which is interpretated as rate of growth in the period t will decline by 15.2% when the growth rate in t-3 is increased by 1%, ceterius paribus.

(b)

testing the null hypothesis which is $\beta _{3}=0$ for growth with a lag of three.ie. $y_{t-3}$ will only give significant results means other coefficients can be taken as zero which is $\beta _{1}=\beta_{2}= 0$ implies that the t period growth depends on the growth 3 years back.

the t test is as explained in part (a) and also the whole regression is significant as from the Fvalue which is 4.288 with a very low pvalue implies that together all the growth rates have a significant impact on growth in period t.