Question 1 (60%)

Consider the following variables , and which follow the univariate processes:

= 0.4 + 0 45.−1 − 0 05.−2 + (Eq.1)

= 2−1 − −2 + (Eq.2)

= 0.1 + 1 7. −1 − 0 8. −2  + 0 1. −3 + (Eq.3)

where , and are all uncorrelated white noise error terms.

a)  Write , and in lag operator notation and show whether they are stationary or non-stationary. For the variable(s) you find to be stationary, compute the long run mean (i.e. ( ), ( ) or ( )).

(20%)

b)  Determine the order of integration of all three variables (hint: for any series which contain a unit root you will need to use differences). If   you wanted to run a meaningful regression using , and how     would you proceed?                                                            (15%)

Data are obtained for the sample period 1984Q1 to 2019Q4 for the quarterly growth rates of the following U.S. variables:

•   All Employees on nonfarm payroll (000s, Seasonally Adjusted)

•   Real Personal Consumption Expenditures (Billions of Chained 2012 Dollars, Seasonally Adjusted)

The employment and PCE growth rates (denoted and from now on) are plotted in Figure 1, below.

Figure 1

c)  Using the Data Excerpt and Regression Output displayed in Appendix 1, make forecasts for and for 2020Q1, 2020Q2, 2020Q3 and       2020Q4 by using the vector autoregressive (VAR) model and the

iterated

multistep

approach .

(25%)

d)  Provide an analysis of the implications of these forecasts for the U.S. economy. Justify whether you believe the forecasts are likely to be  accurate in the current economic climate following the COVID-19      outbreak and suggest ways in which you could improve them.

(10%)

e)  Based on the VAR(1) model, write down two tests for Granger       causality: one for employment growth on consumption growth and vice versa. Using the output in Appendix 1, What do you conclude about Granger causality between employment growth and            consumption growth?                                                          (15%)

f)  Construct multistep forecast intervals for up until 2020Q4 using the 70% and 30% confidence levels and the additional information in the Root Mean Squared Forecast Error (RMSFE) Output in Appendix 1.      Using either a table of multistep forecast intervals or a fan chart,      provide a brief analysis to interpret how the forecast intervals ‘fan    out’ as we approach 2020Q4.                                              (15%)

Question 2 (40%)

Consider the deterministic quadratic trending process:

= 0  + 1 + 2 2  + (Eq.4)

a)  Show that the first difference of is not weakly stationary. Compare this to the case seen in the lectures where we took the difference of a linear deterministic trending process (where 2  = 0).       (15%)

In economics we typically rule out processes which contain both a unit root and a time trend:

= 0  + 1 + −1 +

(Eq.5)

b)  Explain why this process contains a quadratic trend by performing backwards substitution. What is the key difference between this  process and the one in Equation 4?                                      (15%)

You now obtain data from Eurostat for France and Spain real GDP from 1995Q1 through to 2019Q3, which are displayed in Figure 2, below.

Figure 2

c)  Give a brief description of the time series properties of these two

variables. Using your understanding of the concept of cointegration, suggest reasons why France and Spain real GDP may or may not be

cointegrated.

(10%)

The Augmented Dickey Fuller Test Output in Appendix 2 provides the output of a unit root testing procedure run on these variables in levels and first-     differences. The sequential t-testing procedure has been used to determine the number lags at a 5% significance level.

d)  Provide an analysis of this output and determine whether you find    that France and Spain real GDP follow a unit root process. Determine the order of integration of these variables.                         (30%)

e)  Using the Engle Granger Test Output in Appendix 2, determine      whether you find that France and Spain real GDP are cointegrated. Provide an interpretation of this finding.                             (20%)

f)  Eurostat also provides the real GDP data which has been neither       seasonally nor calendar adjusted. Using Figure 3 (below), discuss the seasonal properties of France and Spain real GDP. Compare and        contrast to that of the United Kingdom real GDP in Figure 4.

(10%)

Figure 3

Figure 4

Appendix 1

Data Excerpt

Regression Output

. var e c, lags(1)

Vector autoregression

Sample:  1984q2 - 2019q4

Log likelihood =  -53.36364

FPE            =   .0078638

Det(Sigma_ml)  =   .0072308

Equation           Parms      RMSE

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0.0000

0.0000

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Root Mean Squared Forecast Error (RMSFE) Output