The name is absent

Development of the test

Consider the standard linear regression model

Vt = x'_tβ + e_t,

E(e_t I x_t) = О,

-^e(⅛) = ^σL

1 X о

⅛f∑^σf = σ , t = 1,...,T

'^→∞ t=ι

where x_t is a K v 1 vector containing the t-th observation on K regressors. The null hypothesis is that the model’s parameters,
β and σ², are constant.

The first-order conditions for the least squares estimates of the parameters are

τ

E fa =°’ i = l,---,K + l
t=ι

The first K first-order conditions determine β. The (K + l)-st equation defines σ² to be the maximum-likelihood (rather than
the unbiased) estimator of the error variance.

Hansen’s test statistic for the ith parameter is

1 ^τ

^{Li =} γy. YYα^,

where S_ll is the cumulative first-order condition through period f for the ith parameter, that is,

t

Sit = Y

3=1

and where

T

Vi = Yfif

t=l

The joint test statistic for a change in the model’s parameters is

1 ^τ

L_c = -γS'_tV~¹S_t,
t=ι

where

ft = (fit, ∙ ∙ ∙ , fκ+l,tf,

S_t = (S_u,..., S_κ+ljtf, and

τ
v = Yftfl.
t=ι

Note that the ¾ above are just the diagonal elements of the matrix V.

Parameter stability is rejected if the test statistics Lt and L_c are large. The test statistics are basically averages of the
squared cumulative first-order conditions, Зц. The cumulative first-order condition for the entire sample, S,τ, equals zero by
construction. Intuitively, the cumulative first-order conditions for subsamples ending in period j, j < T, should wander around

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