The Model

The classical regression model

The model estimated by this program is best understood as a generalization of the standard linear regression model that may be written as

 y_t = x_1,t a₁ + x_2,t a₂ +x_3t a₃ +...  + u_t

with y_t as the value of the dependent variable at time t , x_1,t , x_2,t ,... ,x_n,t  as the values of the explanatory variables x₁ , x₂ ,... ,x_n  at time t, a₁ , a₂ ,... ,a_n as the regression coefficients  for these regressors and u_t as a normally i.i.d. distributed disturbance with mean zero and variance σ².

In vector notation, we write the model as

 y_t =  x_t' a + u_t

with x_t'  = ( x_1,t , x_2,t ,... ,x_n,t ) and a' = ( a₁ , a₂ ,... ,a_n ). The regression coefficients a are taken as constants.

The VC regression model

The model considered here generalizes this standard model by introducing the assumption that the regression coefficients may vary over time. Hence a_1,t  denotes the value of the coefficient belonging to the first variable at time t, a_2,t  for the second coefficient, and so forth:

 y_t = x_1,t a_1,t + x_2,t a_2,t +x_3t a_3,t +...  + u_t

 

With a_t' = ( a_1,t , a_2,t ,,.. , a_n,t )  as the vector of regression coefficients at time t and write

(1)         y_t = x_t' a_t + u_t

 

Regarding the coefficients  a, we assume that they change slowly and unsystematically over time in the sense that the expectation of today's coefficients a_t  is equal to yesterday's  state a_t-1 . The change of coefficient i  - it's disturbance - from period t-1 to period t is denoted by v_i,t, . It is assumed to be normally distributed with expectation zero and variance σ_i² . With the vector of disturbances at time t is denoted by v_t = ( v_1,t , v_2,t ,... ,v_n,t )'   we write

(2)      a_t = a_t-1  + v_t

The model (1), (2) generalizes the classical regression model. If the variances of the disturbances in the coefficients ( σ₁²,σ₂²,...,σ_n² ) go to zero, the classical regression model is obtained as a special case.

In order to estimate the coefficients a_t  for the model (1), (2)  the vector y = (y₁,y₂,…,y_T )' of the dependent variable and the matrix X = ( x₁,x₂,…,x_T )' of the explanatory variables must be supplied as an input. The progranm computes the matrix A = ( a₁,a₂,…a_T )', that is, the time-paths of the coefficients. It gives also their estimated standard deviations and confidence bands.

The variances σ² and ( σ₁²,σ₂²,...,σ_n² ) are calculated by a method-of-moments estimator that coincides with the maximum-likelihood estimator for large samples and has a straightforward interpretation in small samples. The matrix of the estimated coefficients  is obtained as the expectation of A given the data X and y:

 = E{ A | X, y }

The program calculates, further, the covariance matrix of the vector a' = vec( A )' = ( a₁', a₂' ,... ,a_T' ), i.e.  E{ (a -â) (a -â)' }. The square roots of the main diagonal elements of this matrix are the standard errors provided in the output file and used to calculate the confidence bands.  If the control "calculate complete covariance matrix" is selected, the covariance matrix will be printed into a separate file.

Descriptive interpretation

The classical regression model is estimated by ordinary least squares: The minimization of the sum of squares

yields the expectations of the coefficients, i.e. â = E{ a | X, y }. The VC estimator  has a similar descriptive interpretation: It is obtained by minimizing the weighted sum of squares

In other words: The coefficients are selected such that the disturbances in the equation u_t  and in the coefficients  v_i,t  are kept as small as possible, where the changes in the coefficients are weighted with the variance ratios σ² /σ_i² . These ratios determine the decomposition and are adjusted during iteration. The progress window depicts them, and the option don't estimate variances minimizes this weighted sum of squares for given variances taken from the input file.

4 Function

4 References