Specification, Estimation and Evaluation

Following Lundbergh and Terasvirta (1998) that proposed a procedure to estimate the STAR-GARCH class model, this paper uses a two-stage estimation strategy which specifies the conditional mean first, and then the conditional variance. I carry out the estimations as a series of single country regressions. The general rule of estimation consists of initially choosing the maximum lag length to specify a linear autoregressive model. There are two general approaches for determining the appro- priate autoregressive order. First, the partial autocorrelation function (PACF), and

second the Akaike information criterion (AIC) and the Schwarz Bayesian information criterion (SBIC). In addition to these criteria, I use a residual autocorrelation test to avoid omitted autocorrelation according to Terasvirta (1994). This test is based on the Ljung Box-Q statistic for the estimated residuals of the selected autoregres- sive model. Second, I test the null hypothesis of linearity of the conditional mean

= 0in equation(2.3.1)against the alternative of a nonlinear STR model according

to Granger and Terasvirta (1993) and Lundbergh and Terasvirta (1998). I use a LM-type test according to Terasvirta (1994) where the null hypothesis is a linear AR model and the alternative is a logistic STR model of order n. As the transition

function is not identified, I approximate equation (2.3.3) using a Taylor expansion

around = 0. Assuming dmwithout loss of generality and setting the transition

variable equal to |ft d st d |, I apply the linearity test for different d values, and

for each value I test the hypothesis = 0. This test assumes constant conditional

variance with an asymptotic 2 _{distribution. However, I use the F-version of the}

test according to Granger and Terasvirta (1993) as it has better small sample prop- erties. If the null hypothesis of linearity is rejected, I select the delay parameter d

for the STR model with the rule of minimum p-value of the F linearity test. Ini- tially, I assume that the transition function is not identified to perform the linearity test. However, I do not test if the model adjust better to the logistic or exponential STR model as I choose the transition function specified in (2.2.10) according to De

Grauwe and Grimaldi (2006).

I estimate the conditional mean assuming that the conditional variance is not time-varying and then test the hypothesis of ARCH effects in the error process using the Engle (1982) LM-test or the McLeod and Li (1983) test. The Engle Lagrange multiplier test is based on the autocorrelation of the squared residuals and it is asymptotically distributed as 2

q when the null hypothesis of no ARCH effects is

it is commonly used as a diagnostic on the standardised squared residuals to test ARCH effects. This test is based on the Ljung-Box test and evaluate if the first p

autocorrelations for the squared residuals are small in magnitude. Under the null hypothesis of no ARCH effects, the test statistic is asymptotically distributed as 2

p.

If there is evidence of heteroskedasticity, I assume that the conditional variance is parametrized as a standard GARCH model. Finally, I estimate the STR-GARCH model specification and run different misspecification tests to check the adequacy of the model. This involve testing against remaining serial dependence in the condi- tional mean and the conditional variance using a LM test for the residuals and the squared standardised errors, respectively.

In addition, I use the BDS statistic (Brock et al., 1996) to test the null hypothesis of independence and identically distributed standardised errors. The BDS test is nonparametric and is derived from the correlation integral which measures the spatial correlation of all possible pairs of m consecutive points. The BDS test statistic is

asymptotically N(0,1) under the null hypothesis. In this paper, I have used the

procedure BDSTEST in RATS to compute the asymptotic values of the test statistic. One of the advantages of this test is that does not require the existence of high-order moments in contrast to most alternative tests. In addition, the BDS test has a high power against a vast class of alternative models if the sample is sufficiently large (i.e. sample sizes of 500 or more). However, one of the disadvantages is that the asymptotic distribution of the test is invariant under some specific conditions when a GARCH process is used to pre-filter the data (de Lima 1996). Moreover, considering the GARCH standardised errors tend to under-reject the null hypothesis due to nuisance parameter effects. De Lima (1996) addresses this issue using a log- transformation of the squared standardised residuals. In addition, this problem may be solved using techniques such as response surface analysis to mitigate the size bias. In this paper, I evaluate the test using the standardised errors. Finally, I use the

likelihood ratio test to evaluate restrictions in the parameters.

It is noteworthy to mention that as in Lundbergh and Terasvirta (1998), I esti- mate all parameters simultaneously. The advantage of using the two-stage estimation procedure is to get appropriate initial values to get convergence in the joint estima- tion of the conditional mean and variance. In addition, I impose the non-negativity constraints in the GARCH model to avoid a negative conditional variance. As in Reitz et al. (2010), I use robust standard errors in my estimations using the BFGS algorithm. This is because the transition function used is a linear transformation of the general transition function suggested by Terasvirta and Anderson (1992) and that affects the assumption of conditional normality in the residuals. Terasvirta (1994) explains the convergence problems derived from nonlinear optimization, and suggest to standardize the transition variable by dividing it by the sample standard deviation for LSTAR models. The STR-GARCH-M model suggested in this paper is consistent with that approach as I scale the transition variable in the logistic func- tion by dividing it by the conditional standard deviation. This is useful for selecting appropriate starting values for the slope parameters 1 and 2 .