BEEM012

University of Exeter Business School

May 2021

APPLIED ECONOMETRICS 2 Module Convenor: Julian Dyer

Duration: 90 minutes + 30 minutes upload time Format: PDF

No word count specified

This is an Open Book Exam Additional Materials needed: None

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Before You Begin

Be sure to follow the instructions carefully, and answer the required number of questions from each section. There are ten pages to this exam.

1. You must answer ONE question in Section A (Time Series Basics – Long Questions)

(1 × 30 marks = 30 )

2. You must answer TWO questions in Section B (Time Series Basics – Short Questions)

(2 × 10 marks = 20 )

3. You must answer TWO questions in Section C (Dynamic Causal Effects)

(2 × 15 marks = 30 )

4. You must answer TWO questions in Section D (Advanced Topics in Time Series)

(2 × 10 marks = 20 )

This adds up to a total of 100 marks.

You have one and a half hours (90 minutes) to write your exam plus a half hour for uploading.

For performing hypothesis tests, there are tables of the necessary critical val- ues on the final page of this exam. All hypothesis tests for this exam are to be performed at the 5% level of statistical significance. Be careful to use the correct critical values for the test you are using.

Under no circumstances may this exam be shared or distributed in any way.

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A TIME SERIES BASICS – LONG QUESTIONS

A Time Series Basics – Long Questions

RECALL: You must answer ONE question from this section! Each question in this section is worth 30 marks.

Question A1

A friend of yours (who has not taken BEEM012!) wants to check that their time series data doesn’t violate the stationarity assumption. They compute the mean at different time periods and are confident that E[Yt] = E[Yt+s] for any t and s, and based on this information alone decide that their time series is stationary.

1. Is your friend correct? Explain why or why not (make sure to include an answer giving the intuition in words!) Make reference to the example of an AR(1) model to explain.

2. Your friend is familiar with the OLS model assumptions, but not with the time series assumptions. Can you explain which of the OLS model assump- tions is most similar to the stationarity assumption?

3. Your friend now realises they aren’t fully confident, and should probably test for a violation of stationarity more formally. Write down the model you will estimate, the null hypothesis you will test and the test statistic you will compute.

4. If they do not reject the null hypothesis, do they need to take any steps to transform their data? Why? If yes, explain the transformation required.

5. You find a test-statistic of -2.4. Do you reject the null hypothesis? Explain why or why not.

Question A2

What test would you use to test for a break in an AR(2) process? Write down the model you would estimate, the null hypothesis and the type of test you would run.

1. Which of the time series assumptions would a break violate? Can you explain which of the OLS model assumptions is most similar to this Time Series assumption?

2. Write down, using the coefficients of the model that you have just described, the intercept of this time series after the break.

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Question A2 A TIME SERIES BASICS – LONG QUESTIONS

3. Similarly, after the break has occurred, if Yt−2 increases by 1, how much of a change in Yt would this lead you to predict?

4. If you aren’t certain when the break might occur, how would you modify the test described above? If you run this revised test and get a test-statistic of 4.75, do you reject the null hypothesis?

5. Can you use the result from this test to conclude how many breaks occurred in your time series? Explain why or why not.

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B TIME SERIES BASICS – SHORT QUESTIONS

B Time Series Basics – Short Questions

RECALL: You must answer TWO questions from this section! Each question in this section is worth 10 marks.

Question B1

For OLS regressions, we can use the Central Limit Theorem to construct our confidence interval. Why can’t we always do this for forecast intervals with Time Series? Refer to the composition of forecasting errors in your explanation.

1. What does the Central Limit Theorem do? How would this help us evaluate the accuracy of our forecasts?

2. If we thought it would be reasonable to assume errors in our time series are normally distributed, would this make constructing forecast intervals easier? Why?

3. Explain one method we explored to empirically estimate the standard devi- ation of our RMSFE.

Question B2

Briefly explain in words the costs and benefits from including additional lags in an autoregression model.

1. Now, write down the formula for the Bayes Information Criterion. Explain which term captures the costs of including more lags, and which term cap- tures the benefits of additional lagged regressors.

2. Let’s say you compute the following values of the BIC and R2 at different p for an AR(p) model:

p BIC(p) R2

1 1.432 0.65

2 1.426 0.68

3 1.391 0.71

4 1.445 0.72

5 1.552 0.75

Write down the autoregression model you would choose as a result.

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Question B3 B TIME SERIES BASICS – SHORT QUESTIONS

Question B3

Look at the plot of a time series in Figure 1. Based on the behaviour of this time series, what problem might be present? Explain how you would formally test for this (just give the name of the test) and if there is a problem, how you would fix it.

Figure 1: A Time Series

Question B4

Explain the concept of a spurious regression with regards to a time series. What form of nonstationarity that we have covered is most likely to cause this problem?

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C PART 2 – DYNAMIC CAUSAL EFFECTS

C Part 2 – Dynamic Causal Effects

RECALL: You must answer TWO questions from this section! Each question in this section is worth 15 marks.

Question C1

Write down the model for an order-2 Distributed Lag model.

1. Explain, in terms of the model coefficients, what the dynamic multipliers are.

2. If we want to understand the cumulative impact of a change in X on Y over the next three periods, how would we compute this from the parameters in this model?

3. Explain intuitively why serial correlation of errors might be present in our original order-3 Distributed Lag model.

4. Explain whether this statement is True or False: “If we estimate our model on data with serially correlated errors, our estimated coefficients will be correct but our standard errors may be incorrect.”

5. Explain the distinction between the assumptions described by the following two error conditions:

i) E[ut|Xt,Xt−1,...] = 0 and

ii) E[ut|...,Xt+1,Xt,Xt−1,...] = 0

6. A friend is asking you how to estimate Dynamic Causal Effects for a project they are working on. Their outcome variable is the price of staple crops which can be consumed or stored, and they want to understand how climate shocks impact prices. In their context, reliable climate forecasts are widely available. Is it reasonable to assume strict exogeneity in this case? Explain why.

Question C2

If we are only able to assume exogeneity, what method do we use to deal with serial correlation of errors when estimating Dynamic Causal Effects? Give the name of the method, as well as an intuitive explanation of how it deals with serial correlation of errors.

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Question C3 C PART 2 – DYNAMIC CAUSAL EFFECTS

1. If we begin with homoskedasticity-only errors

ˆ ?1σv2?

var(β1) = T (σx2)2

how do we adjust to account for serially correlated errors?

2. Explain the issue with estimating the exact value of fT , the correction factor, and discuss the role of the truncation parameter.

3. Write down the formula for f ̃ , the approximated correction factor, using T

the truncation parameter m.

Question C3

If we can assume that our errors follow strict exogeneity, explain how modelling our error as uet = ut − φ1ut−1 allows us to derive models without autocorrelated errors.

1. Explain why this condition is only true under strict exogeneity: E[uet|Xet, Xet−1, . . .] = 0

2. How do we recover our dynamic multipliers from the estimated coefficients from this model?

Yˆ = αˆ + φˆ Y + δˆ X + δˆ X + δˆ X

t 0 1 t−1 0 t (hint: use the expression Yˆ = αˆ + φˆ Y

1 t−1

2 t−2

t−1 0 1 t−2

3. Write the expression for the dynamic multiplier on Yt from being treated one

period ago in terms of these coefficients.

4. Now, consider the second model for estimating dynamic causal effects under strict exogeneity. Briefly sketch the procedure for computing the feasible GLS estimator.

5. Explain the tradeoff between these two methods available under strict exo- geneity.

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+ δˆ X

0 t−1

+ δˆ X + δˆ X

1 t−2 2 t−3

D Part 3 – Advanced Topics

RECALL: You must answer TWO questions from this section! Each question in this section is worth 10 marks.

Question D1

Your friend now wants to predict future values of two variables, but wants to generate a model that is mutually consistent. What type of model would this be? Write down a simple order-2 example of this model for variables Yt and Xt.

1. How would you estimate the coefficients in this model?

2. Your friend now wants to forecast the period T + 2 using the model you have explained to them. What are the two methods for forecasting T + 2 using information from up to period T?

3. Which of these two methods is generally preferable?

4. Your friend now realises that there are likely nonlinear terms in the equation for Xt that they aren’t specifying correctly. Does this change which method you would recommend for predicting T + 2?

Question D2

What does it mean if a Time Series is integrated with order 2, or in other words, Yt is I(2)? Is Yt stationary? What about ∆Yt? What about ∆2Yt?

1. How would you formally test if this is the case?

2. We discussed the Dickey-Fuller Test as the standard way to test for a unit root process. What are the steps for instead computing the more powerful DF-GLS test for a unit root? Explain using the example of a test against the alternative hypothesis that Yt is stationary around an unknown mean.

Question D3

If we have two time series Yt and Xt that both have stochastic trends, what does it mean for them to be cointegrated?

1. Explain how to test for cointegration in the case that the cointegration co- efficient is unkown.

2. What is the error correction term?

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D PART 3 – ADVANCED TOPICS

Question D4 D PART 3 – ADVANCED TOPICS

3. Explain how to use the error correction term to construct a Vector Error Correction Model.

Question D4

If a time series exhibits volatility clustering what does it mean about the vari-

ance of the outcome from one period to the next?

1. ModifythefollowingARCH(2)modeltoarriveattheequationforaGARCH(2,2) model of conditional variance.

σt2 = α0 + α1u2t−1 + α2u2t−2

2. Is the following statement True or False? Be sure to explain your answer, and refer to the terms in your model above!

“The ARCH model allows us to test if a large-magnitude shock in one period makes a large-magnitude shock (positive or negative) more likely in the next period.”

3. How are ARCH and GARCH parameters estimated? (Just give a brief intu- itive explanation)

END OF EXAM

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Appendix: Critical Values for Hypothesis Testing

Use the following tables of critical values in your answers:

t-Test Critical Values with Standard Normal 10% 5% 1%

Two-Sided Test: Reject if |t| > One-Sided Test (>) Reject if t > One-Sided Test (<) Reject if t <

1.64 1.96 2.58

1.28 1.64 2.33 -1.28 -1.64 -2.33

Dickey-Fuller & ADF Critical Values

10% Drift / Intercept Only -2.57 With Trend -3.13

5% 1% -2.88 -3.46 -3.43 -3.99

QLR Critical Values

Number of Restrictions q=1

q=2

q=3

q=4

q=5

10% 5% 1%

7.12 8.68

5.00 5.86 7.78 4.09 4.71 6.02 3.59 4.09 5.12 3.26 3.66 4.53

12.16

F-Test Critical Values

Number of Restrictions m=1

m=2

m=3

m=4

m=5

10% 2.71 2.30 2.08 1.94 1.85

5% 1% 3.84 6.63 3.00 4.61 2.60 3.78 2.37 3.32 2.21 3.02

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