Question 1-10 MCQ

In the model in Question 11, (1) what is the Standard Error of the Estimate? (2) What is the 95% Confidence Interval of the errors? (3) How do you interpret the 95% Confidence Interval of the errors? (4) We get the Lower/Upper 95% Indiv Performance values as follows. How do you interpret this range when Cameras = 7?

\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.

Question 2 [34 marks] A researcher wants to analyse the relationship between the three-month T-bill rate, tb3, the annual inflation rate, inf based on the consumer price index, and the federal budget deficit, def as a percentage of GDP and develops the following model. The data is stored under the file name is tb3.xlsx tb3₂ = a + α₁ inf₂ + a₂ defe + €₂ (a) Use Gretl to obtain a line plot among these three variables (one graph) and comment on the plot. (Hint: to make the graph in Gretl: highlight all three variables right click →→time series plot on a single graph →OK.) [4 marks] (b) Obtain sample correlation coefficient between these variables and comment on the strength of the relationships? (Hint: to obtain correlation in Gretl: → highlight all three variables right click correlation matrix → Ok.) [4 marks] (c) Estimate the above regression model and provide the Gretl output. (Hint: Model ordinary least squares select the appropriate variables to the boxes ➜Ok). [4 marks] (d) Interpret the coefficients a, and a₂. Does the sign of the coefficients agree with your expectations? Explain. [4 marks] (e) Provide and interpret the coefficient of determination, R². [4 marks] (f) Plot the residuals of the model and comment on any pattern. (Gretl: in your regression output window, select the menu Graph →Residual plot →Against Time ok.) [2 marks] (g) Add a one lag of inf and def to the equation in part (a) and re-estimate your model and report the result. Are the coefficients for the two lag variables individually significant at the 5% level? In Gretl: Model ordinary least squares →A dialog box will appear. Choose the "tb3" to the dependent variables box and "inf" and "def" as regressors. From the left bottom corner select →lags→→add 1 lag for "inf" and "def" and then click ok.) [4 marks] (h) Conduct the second order autocorrelation test for the model in part (c) at the 5% significance level. Attach your Gretl results. Clearly states all steps in your test; Null and alternative hypotheses, the auxiliary regression and the test statistic, critical value, your decision and the conclusion. (Gretl: Tests →Autocorrelation Make lags order for test =2→Ok.) [6 marks] (i) Re-estimate your model with Robust standard errors and provide your output. Compare your results with the output in part (c). Comment. (Model →OLS → Choose the "tb3" to the dependent variables box and "inf" and "def" as regressors. From the left bottom corner check the box "Robust standard errors".) [2 marks]

Consider the following regression that examines the wages of graduating economists: wage₁ = 𝛽o + 𝛽1ranki + u¡ where rank is the rank of the school the student graduated from (lower is better) and wage is the wage in dollars per hour. A random sample of recent graduates was taken. a) What is the predicted sign on 𝛽₁? Explain. b) Suppose that the R² of the model is very poor: R² = 0.01. What does this say about the potential to interpret causality in the model? c) Suppose that a government experiment sent students to schools of different ranks: that is, rank; is randomly assigned. Is the estimate of ₁ unbiased? Explain.

When is X NOT a good predictor of Y? Both values in the confidence interval of slope are negative p-value of Hypothesis Test on the Slope is greater than 0.05 Both values in the confidence interval of slope are positive p-value of Hypothesis Test on the Slope is less than 0.05

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.