ANSWER ALL QUESTIONS ON THE EXAM PAPER AND SHOW FULL WORKING Question 1 Heteroskedasticity[10 marks] A researcher on the economics of innovation is interested in the country-level factors that...

ANSWER ALL QUESTIONS ON THE EXAM PAPER AND SHOW FULL WORKING

Question 1
Heteroskedasticity[10 marks]

A researcher on the economics of innovation is interested in the country-level factors that predict R&D (Research and Development) investment.To study this, they take data on 27 OECD countries and regress R&D investment (as a percentage of GDP) against economic output (measured as the log of GDP), and educational attainments (the fraction of young people that have some tertiary education).

The following regression equation is used to model the relationship. Hereydenotes the fraction of expenditure on R&D andandare the GDP and educational variables respectively.

The economist is worried that the error term from this model might be ‘heteroskedastic’.Give a brief definition/description of heteroskedasticity and explain why it is a problem for regression models such as this.

(2 marks)

A plot of the residual terms against the log of real GDP is given below in Figure 1.On the basis of this graph, what would you conclude about the error variance in your model?What problems would it raise for your estimation? Discuss.

Figure 1. Residuals Log GDP per Capita – R&D Model

(2 marks)

Suppose you feel that the error term of the stated model is heteroskedastic and has the structure

Perform a GLS transformation of the modelsuch that the errors will be homoskedastic. State the transformed model.

(3 marks)

Prove theoretically that the errors from the transformed model will have a constant variance.

(3 marks)

Question 2
Autocorrelation[8 marks]

A researcher studying energy markets is trying to produce a model for daily electricity prices () based upon daily temperatures (). A data set containing 201 observations is analyzed where prices are measured in cents per kilowatt-hour while temperatures are in degrees Celsius. The model below is estimated

where both variables are stationary.

The output from Eviews is given in Table 1.

Table 1. Eviews Output for Electricity Model

To visualise the model the researcher also produces the following plot:

Figure 2. Residual, Actual and Fitted Series for Electricity Price Model

Note: Observation numbers ordered by time are given on the horizontal axis. The right vertical axis gives price and the left vertical axis gives the residual.

Based on Figure 2 comment briefly on the performance of the model. Are the standard errors reported in Table 1 likely to be correct? Why or why not?

(2 marks)

A Lagrange Multiplier (LM) test may be used to check for autocorrelation in the residuals of the model given in Table 1. The output of the test is given below in Table 2.

Table 2. Lagrange Multiplier Test for Autocorrelation – Electricity Price Model

Give the test equation, the hypotheses, the test statistic and the appropriate p-value.What order of autocorrelation appears to be present in the residuals? Discuss.

(2 marks)

Another test for autocorrelation incomes from the correlogram.The output is reported in Table 3.

Table 3. Correlogram Q-Statistics of Residuals – Electricity Price Model

Are the results of the correlogram consistent with the results from the LM test? Why or why not? Explain your answer.

(2 marks)

Given the results of the LM test and the correlogram, how could the original equation

be modified to model the autocorrelation more effectively? Give equations for two alternative regression models that could potentially account for any autocorrelation that you have observed.

(2 marks)

Question 3
Dummy Variables[8 marks]

A political scientist is interested in the factors that influence voter preferences. To model the effect of various characteristics of US political candidates upon polling performance, the following equation can be estimated

whereis the candidate’s vote share,is the candidate’s age,is the budget of the campaign andis the number of endorsements the candidate received.

The political scientist feels that there may be structural differences in the regression equations for candidates that do and do not have advanced degrees. LetDdenote a dummy variable that is equal to zero if the candidatedoes nothave an advanced degree; and one if the candidatedoeshave an advanced degree.

Help the political scientist by showing the procedure used to conduct the Chow test. Give the unrestricted and restricted models and the null and alternative hypotheses. What conclusion should the political scientist draw if the null hypothesis is rejected?

(4 marks)

Explain what is meant by the term ‘Dummy Variable Trap’.

(2 marks)

In a famous research paper, David Card and Alan Krueger used a difference in difference estimator to evaluate the effect of minimum wages on employment in the US. A minimum wage was implemented in New Jersey (NJ) but not in Pennsylvania (PA) and the authors took employment data in both states before and after this occurred. Let FTE denote the level of full time equivalent employment,D denote a dummy variable indicating the time period after the wages were introduced, and NJ denote a dummy variable that indicates New Jersey.

Card and Krueger estimated the model

and the output is in Table 4.

Table 4. Difference-in-Difference Equation for FTE Employment

What effect did the minimum wage have on FTE employment in New Jersey? Is this result consistent to what is predicted by economic theory?Why or why not?

(2 marks)

Question 4
Instrumental Variables[9 marks]

Since the early 2000s, the United States has seen a dramatic increase in the abuse of prescription opioids (a strong painkiller chemically derived from heroin). Approximately 500,000 deaths have been attributed to the drug, and its usage has been described as an ongoing emergency for public health.

Consider a health economist who is interested in determining the health effects of opioid use.She takes numerical summary data on the health (y) of individuals (a number from 0-100 where higher values indicate better health), and regresses this against their age (), body mass index (), education (),and a gender dummy (). She also includes the key variable measuring opioid use (). The model is the following linear specification:

Explain to the health economist why variablemay be endogenous withy, and hence whycannot be interpreted as the causal impact of opioid use on health.

(3 marks)

To estimate a causal effect in this context, at least one valid instrument is required. List three statistical properties required of a variable in order to act as a valid instrument.

(3 marks)

Suppose you determine two valid instrumentsandfor estimating the casual impact of opioid use upon health. Explain how the Hausman test employing instrumentsandcan be used to determine if the variableis endogenous. Outline the two-stage testing procedure, including the test equation and hypotheses.

(3 marks)

Question 5
Non-Stationary Time Series[15 marks]

Daily prices for three stock market indices (from Hong Kong - Hang Seng, Japan - Nikkei and the US – S&P500) are shown below.There are approximately 250 observations from each series sourced from 2020-2021.

Figure 3. Value of Hang Seng Index – 2020-2021

Figure 4. Value of Nikkei 225 - 2020-2021

Figure 5. Value of S&P500 – 2020-2021

A financial analyst examines the time-series properties of each variable by estimating the following Dicky-Fuller equations:

(1)

(2)

(3)

To test for stationarity the following hypotheses are used

and tau (τ) statistics are obtained for each variable using the three test equations. The results for the three indices are given in Table 5.

Table 5. τ Statistics for Dickey Fuller Tests – Hang Seng, Nikkei, S&P500

By identifying the appropriate test statistic and critical value for each variable, determine whether the prices of the indices are stationary at the 5% significance level.

(3 marks)

Briefly explain (i.e. one sentence each) how the appropriate test statistics and critical values are determined in each case.

(3 marks)

The analyst feels that due to some regional similarities, the Hang Seng and Nikkei indices might be cointegrated. To test for cointegration they run the following regression:

A plot of the residual seriesis presented below.

Figure 6. Residual Plot – Cointegrating Equation

On the basis of the plot, do you think the Hang Seng and Nikkei indices are cointegrated?What features of the plot would you look for in determining whether the series are cointegrated or not? Explain.

(2 marks)

To test for cointegration the analyst performs a unit root test upon these residuals. If a value ofis obtained, determine if the series are cointegrated. Show your working.

(3 marks)

Suppose the analyst decides that the Nikkei and Hang Seng indices are I(1) and proceeds to model them in first differences. They specify the following ARDL model

whereis the change in the Nikkei in timetandis the change in the Hang Seng at timet. The results are reported below in Table 6.

Table 6. Autoregressive Distributed Lag Model – Nikkei and Hang Seng

Interpret the model by briefly discussing its dynamics.Does the Nikkei index exhibit momentum effects?Do changes in the Hang Seng index appear to drive changes in the Nikkei?If so, how long does it take for effects to spill over from the Hong Kong market to the Japanese market? Use a significance level of 10% to answer this question.

(4 marks)