1.  (40 points) In this question we are going to use the data in the file “dataQ1.xlsx” that I have uploaded on the Brightspace.  These data are simulated, so we know exactly what the true data generating process is. In all sub-points of the question, if you are asked to perform statistical inference you can use the critical values for the N(0, 1) distribution given below.

a. (10 points) In the first exercise, we are going to use the variables y1t  (column B) and xt  (column C). I have generated the variable y1t  as:

y1t = 0.4xt + ut                                                                                         (1)

where the innovation ut  satisfies E(ut) = 0, var(ut) = σu(2)  < ∞ (constant), E(utut−j) = 0 for all j  0, and E(xtut) = 0.

i. Using OLS, run the regression:

y1t = α + βxt + ut

Report the estimated parameters  and , together with the standard errors SE() and SE().    ii.  Test the H0  that the parameter α is equal to its true value given in (1).  Test the H0  that the parameter β is equal to its true value given in (1).  In both tests, use a two-sided alternative

hypothesis. Can you reject the H0 at the 5% significance level? At the 1% significance level? Explain your answers and show your work.

Answer

i. I obtained:  = 0.0002;  = 0.3603; SE() = 0.0322; SE() = 0.0361 (these are homoskedasticity- only standard errors).

ii. In the true model, α = 0, so we test

H0: α = 0    vs.     H1: α  0

t-stat =  = 0.0062

| t-stat | < 1.96 and | t-stat | < 2.576

We fail to reject the H0  at the 5% and 1% significance levels.

In the true model, β = 0.4, so we test

H0: β = 0.4    vs.     H1: β  0.4

t-stat =  = −1.0997

| t-stat | < 1.96 and | t-stat | < 2.576

We fail to reject the H0  at the 5% and 1% significance levels.

b. (15 points) In the second exercise, we are going to use the variables y2t  (column F) and xt  (column G, which is the same as column C).  The variable y2t  was generated using (1), but in this case ut was such that ut  = γzt + vt, where γ > 0 is a parameter, zt  is a variable, and vt  is an innovation term with E(vt) = 0, var(vt) = σv(2)  < ∞ (constant), and E(vtvt−j) = 0 for all j  0. In addition, we

know that E(xtvt−j) = 0 for all j and E(ztvt−j) = 0 for all j .

The variable zt  is such that E(zt) = 0, var(zt) = σz(2)  < ∞, and E(xtzt) > 0.

i.  Write the expressions for E(ut), var(ut) = σu(2) , E(utut−j), and E(xtut).  Explain if all the OLS assumptions hold in this case. Show your work.

ii. Using OLS, run the regression: y2t = α + βxt + ut

Test the H0 that the parameter α is equal to its true value given in (1). Test the H0 that the param- eter β is equal to its true value given in (1). In both tests, use a two-sided alternative hypothesis. Can you reject the two null hypothesis at the 5% significance level?  At the 1% significance level? Explain your answers and show your work.

iii.  Provide an interpretation of your results.  Why do you think the estimated parameters  and  (and their standard errors) are different compared to those that you computed in part a. of the question? (It is not necessary to provide a mathematical proof, a discussion in words is sufficient).

Answer

i. E(ut) = E(γzt+ vt) = γE(zt) + E(vt) = 0

var(ut) = var(γzt+ vt) = γ2var(zt) + var(vt) + 2γcov(zt,vt) = γ 2 σz(2) + σv(2) + 2γcov(zt,vt) cov(zt,vt) = E(ztvt) − E(zt)E(vt) = E(ztvt)

But we are told that E(ztvt−j) = 0 for all j, so cov(zt,vt) = E(ztvt) = 0.

We can finally write: var(ut) = γ 2 σz(2) + σv(2)

E(utut−j) = E[(γzt+ vt)(γzt−j + vt−j)] = E(γ2 ztzt−j + γzt−jvt+ γztvt−j + vtvt−j) = γ2 E(ztzt−j) + γE(zt−jvt) + γE(ztvt−j) + E(vtvt−j)

Again, we are told that E(ztvt−j) = 0 for all j, so E(zt−jvt) = 0 and E(ztvt−j) = 0.  In addition, we know E(vtvt−j) = 0 for all j  0.  However, we don’t have information about E(ztzt−j), so we cannot assume that this measure is equal to zero.

So we have: E(utut−j) = γ2 E(ztzt−j) for all j  0.

E(xtut) = E[xt(γzt+ vt)] = E(γxtzt+ xtvt) = γE(xtzt) + E(xtvt)

We are told E(xtvt−j) = 0 for all j, so E(xtvt) = 0. However, E(xtzt) > 0, so this term is not equal to zero.

We can finally write: E(xtut) = γE(xtzt)

About the OLS assumptions, we have found that E(ut) = 0 and var(ut) = γ 2 σz(2)  + σv(2), which is constant and finite as σz(2)  and σv(2)  are constant and finite. These two assumptions are satisfied.        However, E(utut−j)  0 and E(xtut)  0, so these two assumptions do not hold.

ii. My estimates are now:  = −0.0372;  = 0.6152; SE() = 0.0329; SE() = 0.0369 (again, these are homoskedasticity-only standard errors).

H0: α = 0    vs.     H1: α  0

t-stat =  = −1.1307

| t-stat | < 1.96 and | t-stat | < 2.576

We still fail to reject the H0  at the 5% and 1% significance levels.

H0: β = 0.4    vs.     H1: β  0.4

t-stat =  = 5.8320

| t-stat | > 1.96 and | t-stat | > 2.576

For β, this time we reject the H0  at the 5% and 1% significance levels.

iii.  As we have seen, the assumptions E(utut−j) = 0 (for all j  0) and E(xtut) = 0 do not hold in the true data generating process in this case.  As a consequence, we cannot guarantee that the OLS estimator is unbiased and consistent and that the standard errors that we have computed are correct. In the specific sample that I generated, this issue leads to a situation in which we reject a null hypotheses about the parameter β that is actually true.

c. (15 points) In this last exercise, we are going to use the variables zt  (column H) and wt  (column I). I have generated the variable zt  as:

zt = 10[(wt)0 5.]et                                                                                        (2)

i. Write (2) as the linear regression model:

yt = γ + δxt + ut

where yt  = ln(zt), xt  = ln(wt), and ut  = ln(et). What is the true value of the parameters γ and δ in this linear regression model? Explain your answer.

ii.  Estimate this linear regression model using OLS (note that the data was generated so that all the assumptions of this linear regression model hold).  Test the H0  that the parameter γ is equal to its true value.  Test the H0  that the parameter δ is equal to its true value.  In both tests, use a two-sided alternative hypothesis.  Can you reject the H0  at the 5% significance level?  At the 1% significance level? Explain your answers and show your work.

iii.  Assume that the variable wt  increases by 2 percentage points between time t − 1 and time t. Use your estimated model to predict how the variable zt would change between time t − 1 and time t following this increase. Show your work.

Answer

i. From (2) we have:

ln(zt) = ln(10) + 0.5ln(wt) + ln(et)

yt = ln(10) + 0.5xt + ut

Therefore, γ = ln(10) = 2.3026 and δ = 0.5.

ii.  = 2.3075;  = 0.5195; SE() = 0.0262; SE() = 0.0293 (again, these are homoskedasticity-only standard errors).

H0: γ = 2.3026    vs.     H1: γ  2.3026

t-stat =  = 0.1870

| t-stat | < 1.96 and | t-stat | < 2.575

We fail to reject the H0  at the 5% and 1% significance levels.

H0: δ = 0.5    vs.     H1: δ  0.5

t-stat =  = 0.6655

| t-stat | < 1.96 and | t-stat | < 2.575

We fail to reject the H0  at the 5% and 1% significance levels.

iii. Using the estimated model, we can write:

t = 2.3075 + 0.5195xt

and: t − t −1 = 0.5195(xt− xt−1)

Using the definition of yt  and xt, we have:

l—n(zt) − ln1 ) = 0.5195[ln(wt) − ln(wt−1)]

We know that [ln(wt) − ln(wt−1)] = 2 (i.e., wt  increases by 2 percentage points between time t − 1 and time t). Then our estimated model predicts that zt will increase by 0.5195(2) = 1.039 percentage points between time t − 1 and time t as a consequence of the increase in wt .

2.  (60 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. We are going to use U.S. data. Go to the webpage of FRED (https://fred.stlouisfed.org/) and download the variables listed below (the codes inside the brackets 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 long enough (say, at least 120 months).

❼ Wilshire US RESI (code: WILLRESIPR)

❼ Wilshire 5000 Full Cap Price Index (code: WILL5000PRFC)

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

❼ Unemployment rate (code: UNRATE)

❼ A price index for the US (you can choose the CPI, core CPI, GDP deflator, PCE, core PCE,...)

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

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

❼ returns on the market index, Rm,t  = the percentage change in the Wilshire 5000 Full Cap Price

Index;

❼ risk-free rate, Rf,t  = the 3-Month Treasury Bill:  Secondary Market Rate (no transformation

needed);

❼ excess returns on the real estate index, yt = Rr,t− Rf,t;

❼ excess returns on the market index, xt = Rm,t− Rf,t;

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

❼ the inflation rate πt = the percentage change in your price index (CPI, core CPI, GDP deflator,

PCE, core PCE,...).

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

Answer

As my price index, I used the core PCE (Personal Consumption Expenditures Excluding Food and Energy, PCEPILFE). My data go from Jan.  2000 to Dec.  2019.  You can see my data on the “dataHmw1” excel file.

b.  (12 points) We want to study the relationship between excess returns on the real estate index, yt, and excess returns on the market index, xt, using a version of the CAPM (Capital Asset Pricing Model). The econometric model can be written as:

yt = α + βxt + et                                                                                       (3)

where et  is a innovation term.  Assume that et  follows the normal distribution N(0,σe(2)) and that E(etet−j) = 0 for all j  0. In addition, assume that E(xtet) = 0.

i. First, we want to examine some of the properties of our data on the variables yt and xt . Compute the sample variance of yt, the sample variance of xt, the sample covariance between yt  and xt, and the sample correlation between yt  and xt .

To compute these measures, you can use the formulas available on Excel or in the textbooks or on the “Review of probability and statistics” file. Report the results on your assignment, and show any calculations on the same Excel file in which you have recorded your variables.

ii. Now write the expression for cov(yt,xt) implied by model (3). How does the sign of the parameter β affect the sign of cov(yt,xt) in this model? Based on the sample statistics that you computed in part b.i., what sign do you expect your estimated  to have?  Explain your answer and show your work.

Answer

i.  In my data, the sample variances are sy(2)  = 28.487 and s北(2)  = 18.539, the sample covariance is sy,北 = 16.911, and the sample correlation is ry,北 = 0.736.

ii. cov(yt,xt) = cov(α + βxt+ et,xt) = βcov(xt,xt) + cov(et,xt) = βvar(xt) where the last step follows from the fact that E(xtet) = 0.

Since var(xt) is always a positive number, then the sign of the parameter β determines the sign of cov(yt,xt) in our model.  In my data, I find a positive sample covariance between yt  and xt, so I expect my estimated  to be a positive number.

c. (12 points)

i.   Estimate the linear regression model (3) by OLS and report your estimated parameters and standard errors.

ii.  Assume that the excess return on the market index increases by 5 percentage points between time t − 1 and time t, i.e.  xt − xt−1  = 5.  Use your estimated model to predict the change in the excess return on the real estate index between time t − 1 and time t following this increase.  Show your work.

iii. Compute the 99% confidence interval for the parameter β . Are you 99% confident that the true β is positive (or negative)? Would you reject the H0 that β is equal to zero in a two-sided test with a 1% significance level? Explain your answers and show your work.

Answer

i.  I obtain:   = 0.1080;  = 0.9122; SE() = 0.2425; SE() = 0.0544 (homoskedasticity-only standard errors).

ii. t − t −1 = (xt− xt−1) = 5 = 5(0.9122) = 4.561

The excess return on the real estate index is predicted to increase by about 4.56 percentage points. iii. My interval is: [0.9122 − 2.576(0.0544);0.9122 + 2.576(0.0544)] = [0.7721;1.0523]

My interval only includes positive values, so I am 99% confident that the true value of β is positive. The value β = 0 is not included in my confidence interval. This means that I would reject the H0 that β is equal to zero in a two-sided test with a 1% significance level.

d. (11 points) We want now to examine whether the excess returns on the real estate index are related to two central macroeconomic variables, unemployment and inflation. Consider the following linear regression model:

yt = β 1 + β2ut+ β3 πt + wt                                                                        (4)

where all the variables are as previously defined and wt  is an innovation term.  Assume that wt follows the normal distribution N(0,σw(2)), and that E(wtwt−j) = 0 for all j  0. In addition, assume that E(utwt) = 0 and E(πtwt) = 0.

i.   Estimate the linear regression model (4) by OLS and report your estimated parameters and standard errors.  In addition, compute the residual sum of squares from the estimated model (we will use it in part e. of the question).

ii.  Run individual tests of the H0  that each of the parameters β2  and β3  is equal to zero.  Use a two-sided alternative hypothesis and a 5% significance level.   According to your data, do the unemployment rate and the inflation rate have an effect on the excess returns of the Wilshire US RESI real estate index? Show your work.

Answer

i. β华1 = −6.8295; β华2 = 0.8572; β华3 = 5.6322;

SE(β华1) = 1.2900; SE(β华2) = 0.1844; SE(β华3) = 3.4465 (homoskedasticity-only standard errors). Residual sum of squares = 6211.861. (See the STATA table.)

ii. H0: β2 = 0    vs.     H1: β2  0

t-stat = (0.8572−0)/0.1844 = 4.6486; | t-stat | > 1.96; we reject the H0 at the 5% significance level. H0: β3 = 0    vs.     H1: β3  0

t-stat = (5.6322−0)/3.4465 = 1.6342; | t-stat | < 1.96; we fail to reject the H0 at the 5% significance level.

So in my data the unemployment rate ut  has a statistically significant effect on the excess returns of the Wilshire US RESI real estate index, while the inflation rate πt  does not.

e. (15 points) Finally you want to test the null hypothesis that β2  and β3 in (4) are all 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, you want to use the method of the restricted and unrestricted linear regression model.

i.  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 (RRSS). Report the value of RRSS that you obtained.

ii. Report the residual sum of squares for the unrestricted model (URSS), which you should have already computed in part d. of the question. In addition, indicate the number of restrictions in the test that you want to run (m), your sample size (T), and the number of regressors in the unrestricted model (k).

iii. Compute the value of the F-statistic for your test. Use the critical values below to run the test (your sample size should be large enough, so you can approximate the critical values for the Fm,T−k distribution with those for the Fm,∞ distribution). Do you reject the H0 at the 5% significance level? At the 1% significance level?  Show your work.  Use model (4) to provide an interpretation of the results of your test. What do these results imply in terms of the impact of ut  and πt  on the excess returns on the real estate index, yt? Explain your answer.