# Time Series and Testing Auto Correlation

Consider the following asset pricing model: $$RET_t=0.621+1.414(M_t)+0.732(HML_t)+1.9349(SMB_t)+0.250(RET_{t-1})$$ $$(0.077) \hspace{5mm} (4.141) \hspace{5mm} (3.242) \hspace{5mm} (3.294) \hspace{5mm} (0.011)$$ $$R^2=0.971, \hspace{5mm} SSR=9.319, \hspace{5mm} DW=1.681 \hspace{5mm} \textrm{and} \hspace{5mm}t=1,...,T$$ and $$RET_t=4.631+1.601(M_t)+1.234(HML_t)+1.434(SMB_t)$$ $$(3.977) \hspace{5mm} (2.141) \hspace{5mm} (4.274) \hspace{5mm} (1.535)$$ $$R^2=0.839, \hspace{5mm} SSR=9.319, \hspace{5mm} DW=1.534 \hspace{5mm} \textrm{and} \hspace{5mm}t=1,...,T$$ where the standard errors are given beneath in parentheses, $RET (\%)$ monthly returns of a manager over a ten year period and $M_t, HML$ and $SMB$ are the three Fama factors.

a.) Use specification tests to select the appropriate model to explain returns of the portfolio.

I don't think either model has definitive serial correlation and I think that one model might be a 'special' case of the other. Is this correct? I reckon we should construct an F-test which tests one model against the other coefficient restricted model. Therefore, $H_0: (RET_{t-1})=0$. I am unsure of what $H_1$ is though? How would I then go about conducting the F-test? The F-statistic would be calculated using $$\frac{(R^2_{unrestricted}-R^2_{restricted})/q}{(1-R^2_{unrestricted})/(n-k-1)}.$$ What is the alternative hypothesis? And which values would we plug into the above formula?

b.) Using the model chosen in part a.), test if the factors contribute individually to the returns on this portfolio. Use the relevant hypothesis and statistical tests to evaluate the fit of this model. How was the performance better than the market for the preferred model?

For this part I am not sure if we would test if each variable $(M_t), (HML_t), \textrm{etc})$ including the constant is statistically significant? How would we evaluate the fit of the model?

c.) Using the model chosen in part a.), forecast the portfolio returns $RET_t$ of the manager in the next two months (so that would be $\widehat{RET}_{T+1}$ and $\widehat{RET}_{T+2}$ given at $T$, $RET_T=1\%$. Also, assume that the expected values of the risk factors $(M_t, HML_t, SMB_t)$ is zero in the future. What is the expected long run return to the portfolio?

I am not sure how to begin this part of the question.

Please let me know if I have gone wrong anywhere and help me fill in the gaps.

• Is this a homework exercise? If so, consider adding the 'self-study' tag. Jan 7, 2016 at 8:31
• It's a past exam paper question. Should I still add the 'self-study' tag?
– Levi
Jan 7, 2016 at 13:56
• Well, having familiarized yourself with the tag and what it is used for, it is you who decide. To me it looks like "self study", but I cannot judge. Jan 7, 2016 at 14:19