The assignment is due on Wednesday, March 15th, by 11:59 pm. You are expected to submit: your written answers to the questions in a .pdf ile + the Excel ile with your data for Question 3 + the codes/results from your statistical software printed in a .pdf ile (look for a“Print”command and select the option to “Print to pdf”).  You can collect your .pdf iles together if you prefer.  All the iles are to be submitted through the Brightspace.

Note: it is your responsibility to present the results in an organized way and such that it is clear where the answer to each question is.

1.  (17 points) Consider the following model for the variable yt:

yt = 3 + (L)εt                                                                                          (1)

where εt is a white noise with variance a2 . Assume that the lag polynomial (L) is equal to: (L) = (0.8 + 0.5L + 0.1L2).

a.  (3 points) Write the variable yt  as a function of current and past values of the white noise εt . Show your work. What is the order p of this MA(p) model for the variable yt?

b.   (10 points) Write the expressions for:  u  =  E(yt); V0   =  var(yt); V1   =  cov(yt,yt— 1 ); V2   = cov(yt,yt—2 ); Vj  = cov(yt,yt—j) for j > 2. Show your work.

c.  (4 points) Is the process for the variable yt  described by (1) weakly stationary?  Explain your answer.

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 = 20[Kt(α)H —α)]et                    (2)

with α = 0.75.

a. (10 points) Write (2) as the linear regression model:

yt = β0 + β1 kt+ β2 ht + ut                     (3)

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

i.  According to (2), what are the true values of the parameters β0 , β 1 , and β2  in the linear model

(3)? Explain your answer.

ii.   Show that, regardless of the actual value of α, the original model (2) imposes the following restriction on the parameters of the linear regression: β 1 + β2  = 1.

iii. Estimate (3) using OLS. Report the estimated parameters and their standard errors. In addition, compute the residual sum of squares (call it RSSU) and report its value.

b.  (15 points) We want now to test the restriction β 1 + β2  = 1 using 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 the null hypothesis of interest.  Make sure to deine all the variables that you plan to use in estimating the restricted model.

ii. Estimate the restricted model by OLS and compute the residual sum of squares (call it RSSR). Report the value of RSSR  that you obtained.

iii.  Write the statement of the test.  Indicate the number of restrictions tested (m), your sample size (T), and the number of parameters 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 (the sample size is large, so you can approximate the critical values for the Fm,T —k distribution with those for the Fm,∞ distribution).  Can you reject the H0  at the 5% signiicance level?  At the 1% signiicance level? Show your work.

3.  (58 points) For this question we are going to use U.S. data. Go to 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 quarterly; transform all your variables to quarterly 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 quarters).

The variables are:

• Real Gross Domestic Product (code: GDPC1)

• Real Potential Gross Domestic Product (code: GDPPOT)

• A price index (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:

• the output gap yt, computed as the percentage deviation of Real GDP from Real Potential GDP (you can use the formula yt = 100[ln(GDPC1) − ln(GDPPOT)]);

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

• Tt— 1  = the irst lag of the inlation rate Tt .

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).

b. (10 points) We start by studying the empirical relationship between yt and Tt using the following econometric model:

Tt = α + βyt + et                                                                                       (4)

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

Estimate the linear regression model (4) by OLS and report your results.

Then compute the residuals t  using your estimated  and  and your data for Tt  and yt .  Report these residuals in the Excel ile together with your data.  Show a plot of the residuals.  Do you observe any clear patterns or changes in their dispersion over time?

c. (11 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.  Create the lagged variables t — 1  and t —2  from your residuals t .  Then estimate the auxiliary regression:

t = V0 + V1yt + ρ 1 t — 1 + ρ2 t —2 + ut                                                                 (5)

where all the variables are as previously deined and ut  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 (5). Select the appropriate critical value from the table below to run the test. Can you reject the H0  at the 5% signiicance level? Show your work.

iii.  Interpret the result of this test, together with the plot of the residuals that you produced in part b. of the question. Are you concerned that the assumption E(etet—j) = 0 for all j  0 is not satisied in your data? Explain your answer.

d. (14 points) We decide to include the irst lag of the inlation rate in our model and estimate:

Tt = λ + β1 yt+ β2Tt— 1 + vt                                                                            (6)

where vt  is a random innovation, which is assumed to follow the normal distribution N(0,av(2)) and to satisfy E(vtvt—j) = 0 for all j  0. In addition, assume that E(ytvt) = 0 and E(Tt— 1 vt) = 0.

Estimate the linear regression model (6) by OLS and report your results.

i.   Use a test on the parameter β2  to examine whether the irst lag of the inlation rate has a statistically signiicant impact on its current value Tt . Write the statement of the test with a two- sided alternative hypothesis.  Then run the test using a 1% signiicance level.  Can you reject the H0? Explain your answer.

ii.  Based on the results of the test that you just run, explain whether the assumption E(ytet) = 0 is likely to hold in model (4).  Compute the sample covariance between your yt  and Tt— 1 .  Use the sign of this sample covariance and the sign of your estimate for the parameter β2  in (6) to discuss the sign of the bias potentially afecting your estimate for the parameter β in (4).

iii.  Compare your estimates for the parameter β in (4) and for the parameter β 1  in (6).  Is the diference between these two values what you would have expected based on your discussion about the sign of the bias potentially afecting model (4)? Explain your answer.

e. (13 points) Finally, we want to study the expectation E(Tt).

i.   Write the expression for E(Tt) originating from model  (4).   Write the expression for E(Tt) originating from model (6). Assume that E(Tt— 1 ) = E(Tt). Show all your work.

ii.  Economic theory assumes that in the long run E(yt) = 0 while E(Tt) should be equal to the target inlation rate.  In the U.S. the target inlation rate is around 2%.  Set E(yt) = 0.  Assume

that we want to test the H0  that E(Tt) = 2% using model (4).  Write the statement of the test in terms of the relevant parameter of model (4); use a two-sided alternative hypothesis. Make sure to explain why this is the test that we should be running. Then use your estimates from part b. of the question to run the test. Can you reject the H0  at 1% signiicance level? Show your work.

iii.  Again, set E(yt) = 0.  Assume that we want now to test the H0  that E(Tt) = 2% using model

(6). In this case, we will need to use the method of the restricted and unrestricted linear regression model. Write the statement of the test in terms of the relevant parameters of model (6); use a two- sided alternative hypothesis. Make sure to explain why this is the test that we should be running. Then write the equation of the restricted model that we would need to estimate to run the test. Show your work. (You are not required to actually run the test in this case.)