1.  (75 points) In this question we are going to perform a few exercises using the returns on a stock market index for the real estate sector.  Go to FRED (https://fred.stlouisfed.org/).  Download the variables listed below (the codes are the names of the variables in FRED). The frequency of the data should be monthly; transform all your variables to monthly observations before downloading the data. You can choose the sample period that you prefer, but make sure that it is at least 30 years long (we will split the sample in one of the exercises, so we need a large number of observations).

• Wilshire US RESI (code: WILLRESIND)

• Wilshire 5000 Total Market Full Cap Index (code: WILL5000INDFC)

• 3-Month Treasury Bill: Secondary Market Rate (code: TB3MS)

• Efective Federal Funds Rate (code: FEDFUNDS)

• Unemployment rate (code: UNRATE)

a.  (10 points) Construct your dataset on an Excel ile (which you will submit together with your answers to the assignment). Your ile should include the following variables:

• returns on the real estate index, rr,t  = the percentage change in the Wilshire US RESI;

• returns on the market index, rm,t  = the percentage change in the Wilshire 5000 Total Market Full Cap Index;

• risk-free rate, rf,t  = the 3-Month Treasury Bill:  Secondary Market Rate (no transformation needed);

• excess returns on the real estate index, yt = rr,t − rf,t;

• excess returns on the market index, st = rm,t − rf,t;

• it  = the efective Federal Funds Rate (no transformation needed);

• ut  = the unemployment rate (no transformation needed);

You can use the log diference approximation to compute your percentage changes.  You can also compute the percentage changes directly in FRED before downloading the data. Write your variables so that X% is X in your excel ile (this is the way in which FRED measures the variables; it also simpliies working with these data and interpreting the results of the analysis).

b.  (11 points) We are going to study the relationship between yt  and st  using the baseline version of the CAPM (Capital Asset Pricing Model). The econometric model is:

yt = α + βst + et                                                                                        (1)

where et  is a random innovation, which is assumed to follow the normal distribution N(0,ae(2)) and to satisfy E(etet—j) = 0 for all j  0. In addition, assume that E(stet) = 0.

Estimate the linear regression model (1) by OLS and report your results.

i. Compute the 99% conidence interval for the parameter β . Are you 99% conident that the true β is positive (or negative)? Explain your answer and show your work.

ii. Would you reject the H0  that β is equal to zero in a two-sided test with a 1% signiicance level? Explain your answer.

iii.  Provide an interpretation of your results in terms of the degree of β -risk associated with the Wilshire US RESI Index.

c. (10 points) Compute the residuals t  using your estimated  and , and your data for yt  and st . Report these residuals in the Excel ile together with your data.

i.  Show a plot of the residuals.  Do you observe any clear patterns or changes in their dispersion over time?

ii. Compute the sample autocorrelation between t  and t —j  for j = 1, 2. You can do these compu- tations using your statistical software or Excel. Report your results.

d. (12 points) We want now to examine the possible autocorrelation in the innovation term et  more formally. In order to do so, we are going to run the Breusch & Godfrey’s test for autocorrelation.

i. Estimate the regression:

t = V0 + V1st + ρ 1 t — 1 + ρ2 t —2 + vt                                                                 (2)

where all the variables are as previously deined and vt  is an error term.

Report the R2  of the estimated regression.

ii. Compute the test statistic AR(2) = TAR2 . Make sure to specify TA, the size of the sample that you used to estimate (2). Select the appropriate critical values from the table below to run the test. Do your reject the H0  at the 5% signiicance level? Show your work.

iii.  Interpret the result of this test, together with the information that you obtained in part c.  of the question. Are you concerned about the assumption not being satisied in your model? Explain your answer.

e. (16 points) Next, we want to focus on the assumption of homoskedasticity of the innovation term et . Assume that we suspect that there was a discrete change in ae(2)  within our sample and we want to use the Goldfeld & Quandt’s test to examine this possibility.

i.   Split your sample into two sub-samples T1  and T2, with T1  + T2   = T  (your original sample size).  Select the split date τ so that T1  includes about 1/3 of your observations and T2  about 2/3. Estimate the linear regression model (1) in each of your two sub-samples. Use the estimated models to compute s1(2), the sample variance of t in the irst sub-sample, and s2(2), the sample variance of t in the second sub-sample. To compute these sample variances, use the formula: si(2) = RSSiTi—1, where RSSi is the residual sum of squares in sub-sample i and Ti  is the number of observations in sub-sample i. Show your work.

ii. Our friend Mario argues that we don’t really need to run the Goldfeld & Quandt’s test to check whether there was a structural break in the variance of et . He says that we just need to compare s1(2) and s2(2); if they are diferent from each other, then this clearly implies that the variance of et  in the irst sub-sample is diferent from the variance of et  in the second sub-sample. Explain why Mario is not correct.

iii. Run the Goldfeld & Quandt’s test. Compute the value of the GQ-statistic. You can approximate the critical values for the FT1 —k,T2 —k  distribution with those for the F120,∞    distribution, which are

given below.  (Unfortunately, these critical values are not easily available for all possible values of T1 − k and T2 − k, so we need to approximate them. In our case, the approximation is likely not be too bad as your subsample T1  should include 120-180 observations and your subsample T2  should be much larger than T1). Can you reject the H0? Explain your answer.

iv. Interpret the result of the Goldfeld & Quandt’s test. Are you concerned about the presence of a discrete change in the variance of et  at time τ? Do the results of the test provide information about changes in the variance of et  at times other than τ? Explain your answers.

f.  (16 points) Last, we want to change the focus of our analysis and examine whether the excess returns on the real estate index are related to two main macroeconomic variables, the unemployment rate and the efective Federal Funds Rate (i.e.  the monetary policy interest rate).  Consider the following linear regression model

yt = β 1 + β2 it+ β3ut + wt                                                                             (3)

where all the variables are as previously deined and wt  is a random innovation.  Assume that wt follows the normal distribution N(0,aw(2)), and that E(wtwt—j) = 0 for all j  0. In addition, assume that E(itwt) = 0 and E(utwt) = 0.

We are interested in testing the null hypothesis that β2  and β3  in (3) are both equal to zero at the same time. Formally, the test is:

H0  : β2  = 0  &  β3  = 0       vs.       H1  : β2   0  and/or  β3   0

To run this test, we are going to use the method of the restricted and unrestricted linear regression model.

i.   Estimate the unrestricted linear regression model (3) by OLS and report your estimated pa- rameters and standard errors.  Compute the residual sum of squares (call it RSSU) and report its value.

ii.  Write the equation of the restricted model that you plan to estimate.  Explain why this is the appropriate restricted model to be estimated given your null hypothesis of interest. Then estimate this model by OLS and compute the residual sum of squares (call it RSSR).  Report the value of RSSR  that you obtained.

iii. Indicate the number of restrictions tested (m), your sample size (T), and the number of regressors in your unrestricted model (k). Compute the value of the F-statistic for your test. Use the critical values given below to run the test (your sample size is large enough, so you can approximate the critical values for the Fm,T —k  distribution with those for the Fm,∞    distribution). Do your reject the H0  at the 5% signiicance level? Show your work.

2.  (25 points) In this question we are going to use the data in the ile“dataQ2.xlsx”that I have uploaded on the Brightspace.  These data have been generated using the following model (a Cobb-Douglas

production function):

Yt = 10[Kt(α)H —α)]et

with α = 0.7.

a. (10 points) Write (4) as the linear regression model:

yt = β0 + β1 kt+ β2 ht + ut

(4)

(5)

where yt  = ln(Yt), kt  = ln(Kt), ht  = ln(Ht), and ut  = ln(et). Note that the data was generated so that all the assumptions of this linear regression model hold.