TimeSeriesBook.pdf

The above computation is however not valid because

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The above computation is, however, not valid because the serial corre- lation of the time series was not taken into account. Indeed the estimated autocorrelation function shown in Figure 4.7 clearly shows that the growth rate is indeed subject to high and statistically significant autocorrelations. Taking the Bartlett function as the kernel function, the rule of thumb formula for the lag truncation parameter suggest T = 4. The weights in the
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90 CHAPTER 4. ESTIMATION OF MEAN AND ACF computation of the long-run variance are therefore k ( h/‘ T ) = 1 , h = 0; 3 / 4 , h = ± 1; 2 / 4 , h = ± 2; 1 / 4 , h = ± 3; 0 , | h | ≥ 4 . The corresponding estimate for the long-run variance is therefore given by: ˆ J T = 3 . 0608 1 + 2 3 4 0 . 8287 + 2 2 4 0 . 6019 + 2 1 4 0 . 3727 = 9 . 2783 . Using the long-run variance instead of the simple variance leads to a quite different value of the t-statistic: (1 . 4960 - 1) / p 9 . 2783 / 97 = 1 . 6037. The null hypothesis is thus not rejected at the five percent significance level when the serial correlation of the process is taken into account.
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4.5. EXERCISES 91 4.5 Exercises Exercise 4.5.1. You regress 100 realizations of a stationary stochastic pro- cess { X t } against a constant c . The least-squares estimate of c equals ˆ c = 004 with an estimated standard deviation of ˆ σ c = 0 . 15 . In addition, you have es- timated the autocorrelation function up to order h = 5 and obtained the following values: ˆ ρ (1) = - 0 . 43 , ˆ ρ (2) = 0 . 13 , ˆ ρ (3) = - 0 . 12 , ˆ ρ (4) = 0 . 18 , ˆ ρ (5) = - 0 . 23 . (i) How do you interpret the estimated parameter value of 0.4? (ii) Examine the autocorrelation function. Do you think that { X t } is white noise? (iii) Why is the estimated standard deviation ˆ σ c = 0 . 15 incorrect? (iv) Estimate the long-run variance using the Bartlett kernel. (v) Test the null hypothesis that { X t } is a mean-zero process.
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92 CHAPTER 4. ESTIMATION OF MEAN AND ACF
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Chapter 5 Estimation of ARMA Models The specification and estimation of an ARMA(p,q) model for a given re- alization involves several intermingled steps. First one must determine the orders p and q . Given the orders one can then estimate the parameters φ j , θ j and σ 2 . Finally, the model has to pass several robustness checks in order to be accepted as a valid model. These checks may involve tests of parameter constancy, forecasting performance or tests for the inclusion of additional exogenous variables. This is usually an iterative process in which several models are examined. It is rarely the case that one model imposes itself. All too often, one is confronted in the modeling process with several trade-offs, like simple versus complex models or data fit versus forecasting performance. Finding the right balance among the different dimensions therefore requires some judgement based on experience. We start the discussion by assuming that the orders of the ARMA process is known and the problem just consists in the estimation of the corresponding parameters from a realization of length T . For simplicity, we assume that the data are mean adjusted. We will introduce three estimation methods. The
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