# Consider the following autocorrelation and partial

## Consider the following autocorrelation and partial autocorrelation coefficients estimated using 500 observations from a weakly stationary stochastic process

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Question 3 [25 marks]

You estimate the model

_ = + _ + ,

using quarterly data. Results are reported in Table 1.

Table 1: OLS estimates using 174 observations

Dependent variable: Interest rate

 Variable Coefficient Std. Error t-Statistic Prob. CONSTANT 0.045836 0.113503 0.403833 0.6869 INFLATION 0.204347 0.020596 9.921921 0.0000 OUTPUT GAP 0.022601 0.063539 0.355710 0.7225 R-squared ? Mean dependent var 0.753494 Adjusted R-squared ? S.D. dependent var 1.415286 S.E. of regression 1.114094 Akaike info criterion 3.072648 Sum squared resid 193.6281 Schwarz criterion 3.130552 Log likelihood -241.2755 Hannan-Quinn criter. 3.096162 F-statistic 49.48873 Durbin-Watson stat 0.051860 Prob(F-statistic) 0.000000

a)Are and statistically significant? Explain.

b)Calculate and adjusted using the results reported in Table 1. Discuss the two statistics and the results obtained.

Question 4 [25 marks]

a)Consider the following autocorrelation and partial autocorrelation coefficients estimated using 500 observations from a weakly stationary stochastic process :

 Lag ACF PACF 1 0.307 0.307 2 -0.013 0.264 3 0.086 0.147 4 0.031 0.086 5 -0.019 0.049

3

Which of the autocorrelations are statistically significantly different from 0? Also, using both the Box-Pierce and the Ljung-Box test statistics, test the null hypothesis that the first five autocorrelations are all jointly equal to 0.

b)What process would you tentatively suggest could represent the most appropriate model for the series whose ACF and PACF are presented in part (a)? Explain your answer.

c)Two researchers are asked to estimate an ARMA model for a daily USD/GBP exchange rate return series,denoted . Researcher uses the Bayesian Information Criterion for determining the appropriate model and arrives at an

ARMA(0,1). Research uses Akaike’s Information Criterion which deems an

ARMA(2,0) to be optimal. The estimated models are

: ̂ = 0.38 + 0.10−1

: ̂ = 0.63 + 0.17−1 − 0.09−2

where is a White noiseprocess.

You are given the following data:

 = 0.31, −1 = 0.02, −2 = −0.16 −1 = 0.13, −2 = 0.19 Produce forecasts for=the−0.02,next from both models. 4 days, i.e., for times + 1, + 2, + 3, and + 4,

d)How could you determine whether the models proposed in part (c) are adequate?

e)Suppose that the actual values of the series on days + 1+ 2+ 3, and + 4turned out to be 0.620.19−0.32, and 0.72, respectively. Determine which researcher’s model produced the most accurate forecasts.