# 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 in sandwich form is needed. With the HAC option, it is estimated as

, where is the real-valued kernel function[6], b is the bandwidth parameter, and a is the adjustment factor of small sample degrees of freedom (that is, if the ADJUSTDF option is not specified and otherwise , 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

Parzen

Truncated

Tukey-Hanning

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

Parzen

Truncated

Tukey-Hanning

Let denote each series in , and let denote the corresponding estimates of the autoregressive and innovation variance parameters of the AR(1) model on , , where the AR(1) model is parameterized as with . The and are estimated with the following formulas:

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

Parzen

Truncated

Tukey-Hanning

The and are estimated with the following formulas:

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

Parzen

Truncated

Tukey-Hanning

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

The is estimated with the equation

where 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

where T is the sample size, is the largest integer less than or equal to x, and , r, and c are values specified by BANDWIDTH=SAMPLESIZE(GAMMA=, RATE=, CONSTANT=) options, respectively.

If the PREWHITENING option is specified in the MODEL statement, is prewhitened by the VAR(1) model,

Then is calculated by

[6] The HCCME=0 with CLUSTER option sets .