Homework 1

ECO4185 Financial Econometrics - Winter 2023

1.  (40 points) In this question we are going to use the data in the ile“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 irst 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  satisies E(ut) = 0, var(ut) = au(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% signiicance level? At the 1% signiicance level? Explain your answers and show your work.

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  = Vzt + vt, where V > 0 is a parameter, zt  is a variable, and vt  is an innovation term with E(vt) = 0, var(vt) = av(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) = aX(2)  < ∞, and E(xtzt) > 0.

i.  Write the expressions for E(ut), var(ut) = au(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% signiicance level?  At the 1% signiicance 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 diferent 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 sucient).

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 = V + 6xt + ut

where yt  = ln(zt), xt  = ln(wt), and ut  = ln(et). What is the true value of the parameters V and 6 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 V is equal to its true value.  Test the H0  that the parameter 6 is equal to its true value.  In both tests, use a two-sided alternative hypothesis.  Can you reject the H0  at the 5% signiicance level?  At the 1% signiicance 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.

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).

. Wilshire US RESI (code: WILLRESIPR)

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

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

. Unemployment rate (code: UNRATE)

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

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 Full Cap Price

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, xt = Rm,t − Rf,t;

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

• the inlation rate t = the percentage change in your price index (CPI, core CPI, GDP delator, PCE, core PCE,...).

You can use the log diference 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 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.  (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,ae(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”ile. Report the results on your assignment, and show any calculations on the same Excel ile 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 afect 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.

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% conidence interval for the parameter β . Are you 99% conident 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% signiicance level? Explain your answers and show your work.

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 inlation. Consider the following linear regression model:

yt = β 1 + β2 ut + β3 t + wt                                                                             (4)

where all the variables are as previously deined and wt  is an innovation term.  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(utwt) = 0 and E(twt) = 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).