# **3.3. Modeling the Volatility-GDP Relationship**

To investigate the order of integration, the standard Augmented-Dickey-Fuller (ADF) test is complemented by the Kwiatkowaski-Philips-Schmidt-Shin (KPSS) stationarity test. Due to opposed null hypotheses, the inference from any one test is far more significant if confirmed by the other. Unit root test statistics for realized volatility (Table 2, left panel) unanimously confirm stationarity at the 5% and 1% significance level. This is not surprising, considering that realized volatility is calculated from daily changes ρd and is thus a function of _first differences_ of oil prices. Contrarily, industrial production series exhibit a clear time trend. Table 2 (right panel) presents test statistics for Industrial Production in levels, and first and second differences, while the test in levels allows for a linear time trend and intercept. In levels most national industrial production series possess a unit root, whereas Germany and Japan are found to be trend-stationary. In accordance with these results, further analysis is based on the original realized volatility and once differenced industrial production series (denoted RVm and IPm) in order to enable meaningful regression results.

The RV-IP relationship is modeled as a bivariate vector autoregressive process, which describes the dynamic evolution of industrial production and realized volatility as a function of their common history. The VAR requires no distinction between endogenous and exogenous variables, no arbitrary identification restrictions or any other theoretical a priori assumptions about the nature of the economic relationship. Thus it yields a powerful alternative to a structural simultaneous equation model. As realized volatility and industrial production have roots inside the unit circle, the VAR is stable (stationary).

||**Realized V**<br>|**olatility**<br>|||**Industrial P**|**roduction **|||
|---|---|---|---|---|---|---|---|---|
||ADF|KPSS||ADF|||KPSS||
||||Levels|1st diff.|2nd diff.|Levels|1st diff.|2nd diff.|
|USA|-6.327|0.050|-1.145*|-19.619|-9.430|0.191*|0.117|0.031|
|JPN|-5.439|0.057|-3.683|-4.959|-11.831|0.123|0.030|0.046|
|GER|-5.364|0.051|-4.564|-4.240|-13.006|0.073|0.025|0.046|
|KOR|-6.086|0.063|-1.464*|-26.603|-11.123|0.198*|0.032|0.034|
|IND|-5.374|0.047|-0.300*|-18.753|-12.643|0.151*|0.067|0.020|
|MYS|-5.416|0.046|-2.798*|-29.174|-10.496|0.100|0.034|0.023|
||Crit. values|Crit. values||Critical v|alues||Critical val|ues|
|1%|-3.450|0.739||-3.987|||0.216||
|5%|-2.870|0.463||-3.424|||0.146||
|10%|-2.571|0.347||-3.135|||0.119||

**Table 2.** _Left panel:_ Augmented-Dickey-Fuller (ADF) and Kwiatkowaski-Philips-Schmidt-Shin (KPSS) test statistics for RV in respective countries _Right panel:_ ADF and KPSS test statistics for IP in levels (allowing for a linear time trend), first and second differences. IP series which are suggested to be non-stationary at the 5% significance level are marked with an asterisk (*).

The vector of exogenous variables Xm is modelled as a linear function of its own lags and has following reduced form VAR(q) representation: where Xm = [IPm RVm] _′_ , C is a vector of constants, Φi is a matrix of coefficients and εm a vector of white noise error terms with covariance matrix Σ. Furthermore, subscripts m denote the respective month and q denotes optimal lag length.

|||**Opti**<br>|**mal Lag Length **q<br> <br>||||
|---|---|---|---|---|---|---|
||USA|JPN|GER<br>KOR|IND|MYS|**Table 3.**Optimal lag length|
|AIC|12|13|15<br>5|24|5|qfor theVAR(q)processes,<br>|
|BIC|2|2|2<br>1|13|2|determined by AIC and BIC.|

Optimal lag length is determined by the Akaike or Bayesian Information Criterion, AIC and BIC respectively. The BIC determines the optimal model which minimizes the log mean squared error plus a log penalty term, which increases in the number of regressors K; and hence 7