University | University of London (UOL) |

Subject | EC2020: Elements of Econometrics |

Consider the following regression model

The error term has a zero mean, variance equal to 2=X2i; and E (uiuj ) = 0 for i 6= j: You are given a sample of observations f(Yi; Xi)in i=1. You may treat Xi as being non-stochastic. Clearly annotating your answers:

(a)Derive the OLS estimator of In the presence of heteroskedasticity, the OLS estimator remains unbiased (you are not asked to show this). Derive the variance of the OLS estimator.

(b) Discuss how you can obtain the Best Linear Unbiased Estimator (BLUE) given the heteroskedasticity.

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2. Consider the simple linear regression model

in the presence of a correlation between the error and regressor. The regressor exhibits variability in the sample, i.e., Pn i=1(Xi X) 2 6= 0: Under assumptions of homoskedasticity and the absence of autocorrelation, the IV estimator of that uses the instrument Z has the following (asymptotic) variance (no need to prove this statement)

where rXZ 6= 0 is the sample correlation between X and Z and 2u is the variance of the disturbance term u.

(a) Give the formula for IV (you are not asked to derive it).

(b) Provide at least three factors that will help obtain more precise IV parameter estimates for : In your answer explain why the precision of parameter estimates is important.

(c) Discuss the following statement: “If X is not correlated with u; the best choice of instrument is using the regressor itself.

3. Consider a linear regression model

where the zero mean error “t exhibits autocorrelation of an unknown form. We assume our processes are covariance stationery and exhibit weak dependence. The regressor and error may be assumed to be independent.

(a) (3 marks) Explain what it means to say that f”tg T t=1 is covariance stationery. Provide an intuitive discussion of the requirements and indicate why these requirements are desirable.

(b) Recognizing that the errors exhibit autocorrelation, discuss how we can conduct statistical inference on using the OLS estimator. Specifically, discuss how you can test the hypothesis H0 : = 0:7 against the alternative H1 : < 0:7 using the OLS estimator.

4. Consider the following time series model for

the error that is uncorrelated with anything in the past (white noise).

(a) Show that yt is trend stationary when jj < 1:

(b) Show that yt is difference stationary when = 1:

(c) Discuss the importance of distinguishing between trend stationary and difference stationary processes.

5. (a) Define the term “multicollinearity” and provide a real life example where this problem is likely to occur.

(b) Examine whether the following statements are true or false. Give an explanation.

i) In multiple regression, multicollinearity implies that the least-squares estimators of the coefficients are biased and standard errors invalid.

ii) If the coefficient estimates in an equation have high standard errors, this is evidence of high multicollinearity.

**SECTION B**

Answer three questions from this section.

6. We are interested in explaining the willingness of households to buy ecologically produced apples. We use data where each family was presented with a description of ecologically friendly apples, along with prices (in $) of regular apples (regard) and prices of the hypothetical eco-labeled apple (ecoprc).

The variable we want to explain is the dummy variable, eco buy which equals 1 if the household wants to buy ecologically friendly apples and 0 otherwise. Additional household variables we have are, family income in $1000s, famine, household size, size, years of schooling, Educ, and age.

Using a sample of 660 households, the following results were obtained

The heteroskedasticity robust standard errors are reported in square brackets and the (asymptotic) standard errors are reported in parentheses.

(a)Carefully interpret the estimated coefficient of the price of ecologically friendly apples reported in column “OLS A” and discuss whether the effect is statistically significant. In your answer explain why it is important to use robust standards.

(b) It is argued that using the Probit model is better than using the linear probability model when explaining the binary variable eco buy. Discuss the benefits/drawbacks of using the Probit model when trying to explain a binary variable. In your answer explain what the linear probability model refers to

(c)The Probit model B postulates that where (z) is the standard normal cumulative distribution function. Use the likelihood ratio test, to test the joint significance of the nonprice variables. Clearly indicate the null and alternative hypothesis, the test statistic, and the rejection rule.

(d) In this question we are interested in the marginal effect of the price of ecologically friendly apples using Probit model A holding constant the price of regular apples.

i.Indicate how you can obtain the marginal effect of eco PRC using the probit model.

ii. Unlike in the LPM this marginal effect will not be constant. Discuss what computations you would carry out to obtain the marginal effect of a 0.10$ reduction in ecoprc when evaluated at the mean of our explanatory variables (regard mean equals 0.884$ and ecoprc mean equals 1.082$). You are not expected to use your calculator. The clarity of the computations required is enough

7. Let us consider the following ADL(1,1) model

(a) Provide the short-run and long-run effects of X on Y: Explain the difference between these effects.

(b) Discuss what properties your OLS estimators for the ADL(1,1) parameters will have in the presence of the lagged dependent variable. In particular, (i) are the estimators unbiased and consistent, and (ii) should we use robust standard errors? Provide supportive arguments for your answers.

(c) Show that when you omit the relevant variable Yt1 in the above model, you will get evidence of autocorrelation. Explain the result. Hint: You are expected to reformulate your model as

(d) Discuss how you would proceed to test for the presence of autocorrelation in the model in (b) using the Breusch-Godfrey test. You may assume that under the alternative, ut is (or, can be suitably well approximated by) a station

8. Let us consider the following ADL(1,1) model. Let us assume that the error term has zero mean and that the error is uncorrelated with Yt1; Xt and Xt1:

(a) It is important to distinguish whether the above model is spurious or cointegrating. Explain these concepts clearly and highlight their differences.

(b) Discuss how you can test whether the above specification is spurious (null hypothesis) or cointegrating (alternative) using a standard Augmented Dickey-Fuller test. Clearly indicate the test equation, test statistic, and the rejection rule. The critical value provided by Dickey and Fuller for this test at the 5% level of significance is equal to 2:86.

(c) You are told that there exists an Error Correction Model (ECM) that describes the short-run and long-run dynamics between Yt and Xt : What does this tell you as it relates to your answers in (a) and (b)? Provide the Error Correction Model and interpret the various components of the ECM.

(d) Discuss how the ECM can be estimated using Ordinary Least Squ

9. It is postulated that a reasonable demand-supply model for the wine industry in Australia, under market clearing assumption, would be given by

price of beer relative to CPI, Yt = real per capita disposable income, At = real per capital advertising expenditure, and St = storage cost. CPI is the Consumer Price Index.

The endogenous variables in this model are Q and P w; and the exogenous variables are errors do not exhibit any correlation over time.

(a)Provide the reduced form for P w t

.

(b) The OLS estimation of the demand function, based on annual data from 1955-1975 (T = 20); gave the following results (all variables are in logs and figures in parentheses are t-ratios).

At All the coefficients except that of Y have the wrong signs. The coefficient of P w (price the elasticity of demand, 1) not only has the wrong sign but also appears significant.

Explain why the OLS parameter estimator may give rise to these counter-intuitive results. You are expected to use your results in answer (a) to support your answer.

(c) The supply equation is overidentified. Clearly explain this terminology. What distinguishes overidentification from exact identification and under-identification?

Provide one set of assumptions that would render the supply equation exactly identified.

(d)Discuss how you should estimate the supply equation in light of the overidentified

10. To investigate the relationship between the price of wine and consumption of wine, an economist estimates the following regression using a sample of 32 individuals for one week in 2013:

wine denotes the amount of wine consumed per week in milliliters (a medium glass contains 175ml), and price denotes the average price of a medium glass of wine of a selection of wines during the week in GBP (£). The numbers in parentheses are the standard errors.

(a) Discuss what would happen to the parameter estimate of the slope coefficient if we had measured the amount of wine consumed per week in a number of medium glasses instead of milliliters. Explain your answer.

(b) You are asked to test the hypothesis that the demand for wine has an elasticity equal to 1 against a two-sided alternative, using a 5% level of significance. Clearly stating any assumptions you may need, carry out this test.

(c) Construct a 95% confidence interval for the price elasticity of demand and discuss how this interval can be used to carry out the test in (b).

(d) A famous TV chef suggests in a talk show that the demand for wine is less elastic (i.e., less negative) for people who have eaten at a restaurant during the week, arguing that eating in a restaurant encourages people to drink wine regardless of the price.

To test this theory, the economist defines a dummy variable Di that takes the value 1 if an individual i ate at a restaurant during the week, and 0 otherwise. She obtains the following regression result:

i. How does this regression help in assessing the TV chef’s claim?

ii. Conduct a test that may offer support for the TV chef’s claim. Clearly specify the null and the alternative hypothesis. Explain clearly why your test is effective in answering the question of the internet.

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