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数学代写|Practice Questions for Part 1

这是一篇英国的计量经济数学代写

 

  1. Question 1 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 variance at difffferent time periods and are confifident that var(Yt) = var(Yt+s) for any t and s, and based on this information alone decide that their time series is stationary.

  • 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.
  • Your friend is familiar with the OLS model assumptions, but not with the time series assumptions. Can you explain which of the OLS model assumptions is most similar to the stationarity assumption?
  • Your friend now realises they aren’t fully confifident, and should probably test for a violation of stationarity more formally assuming it is an AR(2) process. Write down the model you will estimate, and the null hypothesis you will test.
  • You fifind a test-statistic of -2.4. Do you reject the null hypothesis?

Explain why or why not.

  • In this case, do you need to make any corrections to your data? If so,how do you manipulate your data?
  1. Question 2 For OLS regressions, we can use the Central Limit Theorem to construct our confifidence interval. Why can’t we always do this for forecast intervals with Time Series? Refer to the composition of forecasting errors in your explanation.
  • What does the Central Limit Theorem tell us about the distribution of OLS prediction errors?
  • How would the CLT help us evaluate the likely region of our forecasts?
  1. Question 3 Explain how you would use a Granger causality test to test whether the two lags of Xt are jointly statistically signifificant in the following

ADL(2,2) model:

Yt = β0 + β1Yt1 + β2Yt2 + δ1Xt1 + δ2Xt2 + ut

Make sure to explain your null hypothesis in terms of these coeffiffifficients. Explain what regression model or models you estimate, and how you construct your test statistic.

  1. Question 4 Write down the formula for the Mean Squared Forecast Error for an AR(2) model. Refer to the terms of this model to explain in words the costs and benefifits from including additional lags in an autoregression model.
  • Now, write down the formula for the Akaike Information Criterion.

Explain which term captures the costs of including more lags, and which term captures the benefifits of additional lagged regressors.

  • Let’s say you compute the following values of the AIC at difffferent p for an AR(p) model:

p   AIC(p)

1    1.432

2   1.426

3   1.391

4   1.445

5   1.552

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

  1. Question 5 If you run the following model, what test can you perform?

Yt = β0 + δYt1 β2∆Yt1 + ut

  • What is δ equal to, in terms of the parameters of an AR(2)?
  • What is the null hypothesis for this test? Use your answer above to give some intuition for the relevance of this hypothesis we test.
  1. Question 6 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?
  1. Question 7 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.

  • Write down, using the coeffiffifficients of the model that you have just described, the intercept of this time series after the break.
  • Similarly, after the break has occurred, if Yt2 increases by 1, how much of a change in Yt would this lead you to predict?
  1. Question 8 Look at the plot of a time series in Fig 1. Based on the behaviour of this time series, what problem might be present? Explain (just give the name) how you would formally test for this problem? If this problem is present, how would you fifix it?

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