Econometrics

Problem 1. Consider the following univariate models with the OLS estimates for 1990Q1- 2012Q4. Note that Y, is the sales of a particular company, D1, D2, D3, and D4 are seasonal dummies, and "Time" is a time trend. a. What are the 3 possible components of a time series? Do these models include all these possible components? Clearly explain. b. Given the above estimates, which model do you prefer for forecasting? Explain with enough reasoning. c. Under what condition(s), your preferred model is correctly specified? Explain clearly. d. Under what condition(s), your preferred model is not correctly specified? How do you improve the specification of the model? Explain clearly.

\text { 1. For the Linear Model } Y-X \beta+c \text {, define } X \text { and } \beta \text { as: } (a) Find the least squares estimator of ß. (b) Find the mean, variance, and distribution of the least squares estimator from part(₂) (c) Find the estimator of o². \text { (d) Give the formula for the }(1-\alpha) \% \text { confidence interval for } \beta_{1} \text {. }

1. Consider the following regression: \text { InGDPpC }=\beta_{0}+\beta_{1} \text { Institutions } 1+u_{1} where the dependent variable is In of GDP per capita, the explanatory variable is a measure of institutional quality (a higher value implies better quality institutions),and the subscript i represents countries. [7 marks] a) Draw a scatter plot that demonstrates how this regression would be biased and explain how your scatter plot demonstrates the bias. For simplicity, assume that there are no other sources of bias when creating your scatter plot. Your scatterplot should be clearly labelled and easy to understand. [2 marks] b) Suppose you estimate the above equation using OLS and the estimated value ofB₁ is 0.23. Interpret this coefficient. [1 mark] c) Now, suppose you have a valid instrumental variable. Do you expect the TSL Sestimate to be greater than, less than, or the same as the OLS estimate of B₁?Explain your answer. [2 marks] d) Explain whether or not panel data would be useful for addressing simultaneous causality bias in this context. [2 marks]

2. For the Linear Model Y-XB+c, define Y, X, ß, and (a) Find the least squares estimator of B. (b) Find the mean, variance, and distribution of your least squares estimator from part (a).1 (c) Find the estimator of o². (d) Find the formulas for SST, SSM, and SSE for this particular model. (e) Give the ANOVA Table for this particular model. (f) Determine the appropriate null and alternative hypotheses for this particular model using the F-test statistic from the ANOVA Table.

Question 1 5 marks A boutique beer brewery produces 2 types of beers, Dark-ale and Light-ale daily with a total cost function: TC = 3QD + QD X QL + 4QL where: QD is the quantity of the Dark-ale beer (in kegs) and Q₁ is the quantity of the Light-ale beer (in kegs). The prices that can be charged are determined by supply and demand forces and are influenced by the quantities of each type of beer according to the following equations: PD = 32 QD + QL for the price (in dollars per keg) of the Dark-ale beer and P₁ = 42+2QD - -QL for the price (in dollars per keg) of the Light-ale beer. The total revenue is given by the equation:TR = PD XQD + PL X QL and the profit given by the equation Profit = TR - TC First, use a substitution of the price variables to express the profit in terms of QD and Q₁ only. Using the method of Lagrange Multipliers find the maximum profit when total production (quantity)is restricted to 192 kegs. Note Qp or Q₁ need not be whole numbers. Question 25 marks A farmer discovers that his land has been targeted as a chemical dumping ground with a chemical that is dangerous for growing any crops. It is known that the chemical concentration decays according to the exponential decay process. At the time of discovery, the concentration of the chemical was 15% of the original. One week later, the chemical content reduced to 14%. The police have two suspects, who were both in prison for 15 weeks each at different times for other offences but providing them with alibis (proof of innocence). Suspect A served his sentence ending 35 weeks before the time of the discovery and Suspect B was released from prison 40 weeks before the time of the discovery. Use the exponential decay model to determine whether any of the suspects are innocent.

8. Use the following table to answer the subsequent questions: [5 marks] a) Suppose group 1 is treated by a quasi experiment by 2020, while group 2 is not.Calculate the difference-in-differences estimator. [2 marks] b) Suppose you have the following additional data Does this change your opinion on whether or not the treatment had an effect?Explain your answer. [3 marks]

THE REGRESSION ANALYSIS 1 The Simple Regression Model [30 pts] 1.1 Regression to the Origin [6 pts] Consider the estimation of the following model using a random sample {(y₁, ₁): i=1,2,...,n}, where B₁ is estimated using ordinary least squares (OLS). a. Derive 3₁ using OLS. [1 pt] b. Show under the necessary assumptions that 3₁ is unbiased if the population regression function is y₁ = B₁+U₂. [2 pts] c. Show -under the same assumptions used above that B₁ is biased if the population regression function is Yi = Bo + B₁zi + Ui, with Bo 0. [1 pt] d. Show that 1.2 Regression to a Constant [4 pts] Consider the estimation of the following model using a random sample {(yi): i = 1,2,...,n}. a. Derive Bo using OLS. [2 pts] b. Provide an interpretation of o. [2 pts] Yi = Bo + ûi 1.3 Regression on a Binary Explanatory Variable and Average Treatment Effect [20 pts] Let y be any response variable and a binary explanatory variable. Let {(ri, Yi) : i = 1,2,...,n} be a sample of size n. Let no be the number of observations with ; = 0 and n₁ the number of observations with ₂ = 1. Let Yo be the average of the y, with = 0, and ₁ the average with z; = 1. a - Explain why we can write b. Argue that c. Show that the average of y, in the entire sample, y, can be written as a weighted average: y = (1-I)yo+zÿ₁. d. Show that when z is binary, e. Show that f. Use parts d. and e. to show g. Show, also,

Consider the following regression: Yi = 𝛽o + 𝛽1xi + ui where u; = xe, E(e) = 0, and x and e are independent. a) Show that E(ulx) = 0. b) Show that var (u|x) = x² var(e). c) Since the variance of the error term varies with x, which of the SLR assumptions is violated? Explain. d) Because this SLR assumption is violated, is the estimation of ₁ necessarily biased? Explain.

Based on the model below, which of the following is an example of data extrapolation?

a. Present the OLS SRF estimates. b. Use the Ljung-Box test to check for an autocorrelation problem up to the 3rd-order. Include in your answer the test equation, the null and alternative hypothesis, and your conclusion. c. Use the White test to check for a heteroscedasticity problem. Include in your answer the test equation, the null and alternative hypothesis, and your conclusion. d. Use the cusum of squares test to check for parameter stability? Include in your answer the null and alternative hypothesis and your conclusion. e. Do you suspect a high multicollinearity problem? Explain by presenting your results f. Is the normality assumption empirically valid? Include in your answer the null and alternative hypothesis and your conclusion. g. Is the interest rate function a dynamic model? Include in your answer the null and alternative hypothesis and your conclusion. h. Is the interest rate function dynamically stable? Include in your answer the null and alternative hypothesis and your conclusion i. Test the null hypothesis that (all else remaining equal), inflation expectations do not explain the behavior of the interest rate. Include in your answer the null and alternative hypothesis and your conclusion. j. Find and interpret the short- and long-run effects of a change in inflation expectations on the interest rate. Draw the pattern of the effects over quarters. k. Interpret the coefficient estimate on F-1 (the lagged dependent variable). 1. Are you satisfied with the estimates of the dynamic model? Explain clearly.