The PANEL Procedure

Heteroscedasticity- and Autocorrelation-Consistent Covariance Matrices

The HAC option in the MODEL statement selects the type of heteroscedasticity- and autocorrelation-consistent covariance matrix. As with the HCCME option, an estimator of the middle expression $\Lambda$ in sandwich form is needed. With the HAC option, it is estimated as

$\Lambda _{\mr {HAC}}=a\sum _{i = 1} ^{N} \sum _{t=1}^{T_ i} \hat{\epsilon }_{it} ^{2}\mb {x} _{it} \mb {x} _{it} ^{} +a\sum _{i = 1} ^{N} \sum _{t=1}^{T_ i} \sum _{s=1}^{t-1} k(\frac{s-t}{b})\hat{\epsilon }_{it}\hat{\epsilon }_{is}\left(\mb {x} _{it} \mb {x} _{is} ^{}+\mb {x} _{is} \mb {x} _{it} ^{}\right)$

, where $k(.)$ is the real-valued kernel function [5], $b$ is the bandwidth parameter, and $a$ is the adjustment factor of small sample degrees of freedom (that is, $a=1$ if the ADJUSTDF option is not specified and otherwise $a=NT/(NT-k)$ , where $k$ is the number of parameters including dummy variables). The types of kernel functions are listed in Table 20.1.

Table 20.1: Kernel Functions

Kernel Name	Equation
Bartlett	$k(x)=\left\{ \begin{array}{ll} 1-\|x\| & \|x\|\leq 1 \\ 0 & \text {otherwise} \end{array} \right.$
Parzen	$k(x)=\left\{ \begin{array}{ll} 1-6x^2+6\|x\|^3 & 0\leq \|x\| \leq 1/2 \\ 2(1-\|x\|)^3 & 1/2 \leq \|x\| \leq 1 \\ 0 & \text {otherwise} \end{array} \right.$
Quadratic spectral	$k(x)=\frac{25}{12\pi ^2x^2} \left( \frac{\sin {(6\pi x/5)}}{6\pi x/5} - \cos {(6\pi x/5)} \right)$
Truncated	$k(x)=\left\{ \begin{array}{ll} 1 & \|x\|\leq 1 \\ 0 & \text {otherwise} \end{array} \right.$
Tukey-Hanning	$k(x)=\left\{ \begin{array}{ll} \left(1+\cos {(\pi x)}\right)/2 & \|x\|\leq 1 \\ 0 & \text {otherwise} \end{array} \right.$

When the BANDWIDTH=ANDREWS option is specified, the bandwidth parameter is estimated as shown in Table 20.2.

Table 20.2: Bandwidth Parameter Estimation

Kernel Name	Bandwidth Parameter
Bartlett	$b = 1.1447(\alpha (1)T)^{1/3}$
Parzen	$b = 2.6614(\alpha (2)T)^{1/5}$
Quadratic spectral	$b = 1.3221(\alpha (2)T)^{1/5}$
Truncated	$b = 0.6611(\alpha (2)T)^{1/5}$
Tukey-Hanning	$b = 1.7462(\alpha (2)T)^{1/5}$

Let $\{ g_{ait}\}$ denote each series in $\{ g_{it}=\hat{\epsilon }_{it} \mb {x} _{it}\}$ , and let $(\rho _ a,\sigma _ a^2)$ denote the corresponding estimates of the autoregressive and innovation variance parameters of the AR(1) model on $\{ g_{ait}\}$ , $a=1,...,k$ , where the AR(1) model is parameterized as $g_{ait}=\rho g_{ait-1} + \epsilon _{ait}$ with $Var(\epsilon _{ait})=\sigma _ a^2$ . The $\alpha (1)$ and $\alpha (2)$ are estimated with the following formulas:

$\alpha (1) = \frac{\sum _{a=1}^ k{\frac{4\rho _ a^{2}\sigma _ a^4}{(1-\rho _ a)^6(1+\rho _ a)^2}}}{\sum _{a=1}^ k{\frac{\sigma _ a^4}{(1-\rho _ a)^4}}} \\ \alpha (2) = \frac{\sum _{a=1}^ k{\frac{4\rho _ a^{2}\sigma _ a^4}{(1-\rho _ a)^8}}}{\sum _{a=1}^ k{\frac{\sigma _ a^4}{(1-\rho _ a)^4}}}$

When you specify BANDWIDTH=NEWEYWEST94, according to Newey and West (1994) the bandwidth parameter is estimated as shown in Table 20.3.

Table 20.3: Bandwidth Parameter Estimation

Kernel Name	Bandwidth Parameter
Bartlett	$b = 1.1447(\{ s_1/s_0\} ^2T)^{1/3}$
Parzen	$b = 2.6614(\{ s_1/s_0\} ^2T)^{1/5}$
Quadratic spectral	$b = 1.3221(\{ s_1/s_0\} ^2T)^{1/5}$
Truncated	$b = 0.6611(\{ s_1/s_0\} ^2T)^{1/5}$
Tukey-Hanning	$b = 1.7462(\{ s_1/s_0\} ^2T)^{1/5}$

The $s_1$ and $s_0$ are estimated with the following formulas:

$s_1 = 2\sum _{j=1}^ n{j\sigma _ j} \\ s_0 = \sigma _0+2\sum _{j=1}^ n{\sigma _ j}$

where $n$ is the lag selection parameter and is determined by kernels, as listed in Table 20.4.

Table 20.4: Lag Selection Parameter Estimation

Kernel Name	Lag Selection Parameter
Bartlett	$n = c(T/100)^{2/9}$
Parzen	$n = c(T/100)^{4/25}$
Quadratic Spectral	$n = c(T/100)^{2/25}$
Truncated	$n = c(T/100)^{1/5}$
Tukey-Hanning	$n = c(T/100)^{1/5}$

The $c$ in Table 20.4 is specified by the C= option; by default, C=12.

The $\sigma _ j$ is estimated with the equation

$\sigma _ j = T^{-1}\sum _{t=j+1}^{T}{\left(\sum _{a=i}^ k{g_{at}}\sum _{a=i}^ k{g_{at-j}}\right)}, j=0, ..., n$

where $g_{at}$ is the same as in the Andrews method and $i$ is 1 if the NOINT option in the MODEL statement is specified, and 2 otherwise.

When you specify BANDWIDTH=SAMPLESIZE, the bandwidth parameter is estimated with the equation

$b = \left\{ \begin{array}{ l l } \left\lfloor {\gamma T^{r} + c} \right\rfloor & \text {if BANDWIDTH=SAMPLESIZE(INT) option is specified} \\ \gamma T^{r} + c & \text {otherwise} \end{array} \right.$

where $T$ is the sample size, $\left\lfloor {x} \right\rfloor$ is the largest integer less than or equal to $x$ , and $\gamma$ , $r$ , and $c$ are values specified by BANDWIDTH=SAMPLESIZE(GAMMA=, RATE=, CONSTANT=) options, respectively.

If the PREWHITENING option is specified in the MODEL statement, $g_{it}$ is prewhitened by the VAR(1) model,

$g_{it} = A_{i} g_{i,t-1} + w_{it}$

Then $\Lambda _{\mr {HAC}}$ is calculated by

$\Lambda _{\mr {HAC}}=a\sum _{i = 1} ^{N}\left\{ ((I-A_{i})^{-1})’\left(\sum _{t=1}^{T_{i}}{w_{it} w_{it}’}+\sum _{t=1}^{T_{i}}{\sum _{s=1}^{t-1}{k(\frac{s-t}{b})\left(w_{it} w_{is}’ + w_{is} w_{it}’\right)}}\right)(I-A_{i})^{-1}\right\}$

^[5]The HCCME=0 with CLUSTER option sets $k(.)=1$ .