This example shows how you can use the SSM procedure to specify and fit the so-called dynamic panel model, which is commonly used to analyze a panel of time series. Suppose that a panel of time series follows the model
where t denotes the time index (for example, ); i denotes the panel index (for example, ); is the autoregression coefficient; denote the panel-specific intercepts; are observations on a regression variable with regression coefficient (the same for all panels); are unobserved, random time effects; and are the observation errors. The sequences and are assumed to be independent, zero-mean Gaussian variables with variances and , respectively. This is an example of a dynamic panel model that contains one regressor variable. It is easy to formulate this model equation as a state equation with state of size P—the number of panels. Taking , it is easy to see that the states evolve according to the equation
where (a P-dimensional, diagonal matrix with all its diagonal elements equal to ); is a -dimensional matrix (in a block form) of state regression variables, where the first block is a column that includes all the values that are associated with a given time index (t) and the second block is a P-dimensional identity matrix; is the -dimensional column vector of regression coefficients; and is a P-dimensional column vector of all the disturbances that are associated with time index t. Because and are independent, the covariance matrix of —for example, —is easy to calculate: . This formulation can be easily extended to multiple regression variables, such as variables, by appropriately modifying the term that is associated with the state regression variables—: the new matrix becomes -dimensional and the new regression vector becomes -dimensional.
The cross-sectional data, Cigar
, that are used in the section Getting Started: SSM Procedure are reused in this example. In order to use the SSM procedure to perform the dynamic panel model–based analysis, the input
data set must be reorganized so that it contains the variables that form the -dimensional matrix . For the Cigar
data, the number of panels (the number of regions considered in the study), and the number of regression variables . Therefore, the input data set needs to be augmented by variables that constitute the matrix —the first -dimensional block contains the values of the three regression variables, lprice
, lndi
, and lpimin
, at a given time index (a particular year in this case). The following DATA steps accomplish this task in two steps. In the
first step, the raw data that form the rows of the Cigar
data set are read into a temporary data set, Tmp
, such that all 6*46 = 276 values that are associated with a given year (values of six variables—year
, region
, lsales
, lprice
, lndi
, and lpimin
for 46 panels in a given year) are read in a single row that consists of 276 columns. In the second step, the final input
data set is formed by rearranging Tmp
so that it contains the necessary variables in the proper order—year
(the time index), region
(the panel index), lsales
(the response variable), and the variables that form the -dimensional matrix (w1
, . . ., w2254
).
data Tmp; input u1-u276; datalines; 63 1 4.54223 3.35341 7.3514 3.26194 63 2 4.82831 3.17388 7.5729 3.21487 63 3 4.63860 3.29584 7.3000 3.25037 ... more lines ...
data cigar(keep=year region lsales w1-w2254); array wmat{46, 49} w1-w2254; array ivar{46, 6} u1-u276; set tmp; year = intnx( 'year', '1jan63'd, u1-63 ); format year year.; do i=1 to 46; region = ivar[i, 2]; lsales = ivar[i, 3]; do j=1 to 46; do k=1 to 49; wmat[j,k] = 0; if k = j+3 then wmat[j,k] = 1; if k=1 then wmat[j,k] = ivar[j, 4]; if k=2 then wmat[j,k] = ivar[j, 5]; if k=3 then wmat[j,k] = ivar[j, 6]; end; end; output; end; run;
The following statements specify and fit the dynamic panel model:
proc ssm data=Cigar opt(tech=dbldog maxiter=75); id year interval=year; parms rho / lower=-0.9999 upper=0.9999; parms sigma0 sigma1 / lower=1.e-8; array RegionArray{46} region1-region46; do i=1 to 46; RegionArray[i] = (region=i); end; array cov{46,46}; do i=1 to 46; do j=1 to 46; if(i=j) then cov[i,j] = sigma0 + sigma1; else cov[i,j] = sigma1; end; end; state panelState(46) T(I)=(rho) W(g)=(w1-w2254) cov(g)=(cov) a1(46) checkbreak; comp dynPanel = (RegionArray)*panelState; model lsales = dynPanel; output out=for1 press; run;
The estimates of the regression coefficients and the regional intercepts, which are all statistically significant, are shown
in Output 27.11.1. In particular, the estimated coefficients of lprice
, lndi
, and lpimin
, are –0.26, 0.13, and 0.07, respectively.
Output 27.11.1: Estimates of , , and the Regional Intercepts
Estimate of the State Equation Regression Vector | |||||
---|---|---|---|---|---|
State | Element Index | Estimate | Standard Error |
t Value | Pr > |t| |
panelState | 1 | -0.2627 | 0.0178 | -14.79 | <.0001 |
panelState | 2 | 0.1340 | 0.0130 | 10.30 | <.0001 |
panelState | 3 | 0.0748 | 0.0198 | 3.78 | 0.0002 |
panelState | 4 | 0.4265 | 0.0581 | 7.35 | <.0001 |
panelState | 5 | 0.3825 | 0.0605 | 6.32 | <.0001 |
panelState | 6 | 0.4425 | 0.0582 | 7.61 | <.0001 |
panelState | 7 | 0.3471 | 0.0631 | 5.50 | <.0001 |
panelState | 8 | 0.3686 | 0.0635 | 5.81 | <.0001 |
panelState | 9 | 0.4357 | 0.0614 | 7.10 | <.0001 |
panelState | 10 | 0.3753 | 0.0655 | 5.73 | <.0001 |
panelState | 11 | 0.4249 | 0.0606 | 7.01 | <.0001 |
panelState | 12 | 0.4185 | 0.0604 | 6.92 | <.0001 |
panelState | 13 | 0.3824 | 0.0602 | 6.35 | <.0001 |
panelState | 14 | 0.3942 | 0.0644 | 6.12 | <.0001 |
panelState | 15 | 0.4154 | 0.0626 | 6.64 | <.0001 |
panelState | 16 | 0.3961 | 0.0610 | 6.49 | <.0001 |
panelState | 17 | 0.3765 | 0.0618 | 6.10 | <.0001 |
panelState | 18 | 0.4528 | 0.0608 | 7.44 | <.0001 |
panelState | 19 | 0.4316 | 0.0586 | 7.36 | <.0001 |
panelState | 20 | 0.4357 | 0.0601 | 7.25 | <.0001 |
panelState | 21 | 0.3771 | 0.0639 | 5.90 | <.0001 |
panelState | 22 | 0.3939 | 0.0629 | 6.26 | <.0001 |
panelState | 23 | 0.4122 | 0.0621 | 6.64 | <.0001 |
panelState | 24 | 0.3949 | 0.0605 | 6.52 | <.0001 |
panelState | 25 | 0.4386 | 0.0565 | 7.77 | <.0001 |
panelState | 26 | 0.4118 | 0.0627 | 6.57 | <.0001 |
panelState | 27 | 0.3898 | 0.0604 | 6.45 | <.0001 |
panelState | 28 | 0.3818 | 0.0613 | 6.23 | <.0001 |
panelState | 29 | 0.4343 | 0.0632 | 6.87 | <.0001 |
panelState | 30 | 0.4619 | 0.0625 | 7.39 | <.0001 |
panelState | 31 | 0.3730 | 0.0636 | 5.86 | <.0001 |
panelState | 32 | 0.3784 | 0.0589 | 6.43 | <.0001 |
panelState | 33 | 0.3825 | 0.0625 | 6.12 | <.0001 |
panelState | 34 | 0.3784 | 0.0598 | 6.32 | <.0001 |
panelState | 35 | 0.4093 | 0.0628 | 6.52 | <.0001 |
panelState | 36 | 0.4155 | 0.0597 | 6.96 | <.0001 |
panelState | 37 | 0.3960 | 0.0615 | 6.44 | <.0001 |
panelState | 38 | 0.4075 | 0.0602 | 6.77 | <.0001 |
panelState | 39 | 0.4045 | 0.0586 | 6.91 | <.0001 |
panelState | 40 | 0.3918 | 0.0599 | 6.55 | <.0001 |
panelState | 41 | 0.4350 | 0.0608 | 7.16 | <.0001 |
panelState | 42 | 0.4007 | 0.0602 | 6.65 | <.0001 |
panelState | 43 | 0.3196 | 0.0597 | 5.36 | <.0001 |
panelState | 44 | 0.4337 | 0.0609 | 7.12 | <.0001 |
panelState | 45 | 0.3790 | 0.0634 | 5.98 | <.0001 |
panelState | 46 | 0.3767 | 0.0618 | 6.10 | <.0001 |
panelState | 47 | 0.4392 | 0.0597 | 7.36 | <.0001 |
panelState | 48 | 0.3932 | 0.0603 | 6.51 | <.0001 |
panelState | 49 | 0.3938 | 0.0616 | 6.40 | <.0001 |
Output 27.11.2 shows the estimates of the autoregression coefficient , the observation error variance , and the variance of the time effect (variance of ) .
Finally, you can compare the fit of the dynamic panel model with the fit of the model that is discussed in the section Getting Started: SSM Procedure. Output 27.11.3 shows the likelihood-based information criteria for the dynamic panel model, and Output 27.11.4 shows the same information for the other model.
Similarly, Output 27.11.5 shows fit criteria based on the delete-one cross validation error for the dynamic panel model, and Output 27.11.6 shows the same information for the other model.
On the basis of both these considerations, the dynamic panel model appears to provide a better fit for the Cigar
data than the model that is fit in the section Getting Started: SSM Procedure.