How do I replicate these simple state space models?

[Michael Anbar]

I'm working through this book (Commandeur, Jacques JF, and Siem Jan Koopman. An introduction to state space time series analysis. Oxford University Press, 2007.) and I want to replicate a few of the simple models in Stata 13.1. The two related models I'm working on now are special cases of the local level model:

\begin{align}
y_t &= u_t + \epsilon_t \\
u_{t+1} &= u_t + \xi_t
\end{align}

In the first example (the "deterministic level"), $$\xi_t = 0$$, so $$y_t = u_1 + \epsilon_t$$. I know this is a trivial model to estimate (because we're just estimating the mean and variance of the y's), but I wanted to match the book and estimate it with state space commands. Here is the code I'm using for this example (this code includes the data; the actual sspace statement is at the bottom):

clear
cls

input ksi t
     ksi     t  
    1687   108  
    1508   109  
    1507   110  
    1385   111  
    1632   112  
    1511   113  
    1559   114  
    1630   115  
    1579   116  
    1653   117  
    2152   118  
    2148   119  
    1752   120  
    1765   121  
    1717   122  
    1558   123  
    1575   124  
    1520   125  
    1805   126  
    1800   127  
    1719   128  
    2008   129  
    2242   130  
    2478   131  
    2030   132  
    1655   133  
    1693   134  
    1623   135  
    1805   136  
    1746   137  
    1795   138  
    1926   139  
    1619   140  
    1992   141  
    2233   142  
    2192   143  
    2080   144  
    1768   145  
    1835   146  
    1569   147  
    1976   148  
    1853   149  
    1965   150  
    1689   151  
    1778   152  
    1976   153  
    2397   154  
    2654   155  
    2097   156  
    1963   157  
    1677   158  
    1941   159  
    2003   160  
    1813   161  
    2012   162  
    1912   163  
    2084   164  
    2080   165  
    2118   166  
    2150   167  
    1608   168  
    1503   169  
    1548   170  
    1382   171  
    1731   172  
    1798   173  
    1779   174  
    1887   175  
    2004   176  
    2077   177  
    2092   178  
    2051   179  
    1577   180  
    1356   181  
    1652   182  
    1382   183  
    1519   184  
    1421   185  
    1442   186  
    1543   187  
    1656   188  
    1561   189  
    1905   190  
    2199   191  
    1473   192  
    1655   193  
    1407   194  
    1395   195  
    1530   196  
    1309   197  
    1526   198  
    1327   199  
    1627   200  
    1748   201  
    1958   202  
    2274   203  
    1648   204  
    1401   205  
    1411   206  
    1403   207  
    1394   208  
    1520   209  
    1528   210  
    1643   211  
    1515   212  
    1685   213  
    2000   214  
    2215   215  
    1956   216  
    1462   217  
    1563   218  
    1459   219  
    1446   220  
    1622   221  
    1657   222  
    1638   223  
    1643   224  
    1683   225  
    2050   226  
    2262   227  
    1813   228  
    1445   229  
    1762   230  
    1461   231  
    1556   232  
    1431   233  
    1427   234  
    1554   235  
    1645   236  
    1653   237  
    2016   238  
    2207   239  
    1665   240  
    1361   241  
    1506   242  
    1360   243  
    1453   244  
    1522   245  
    1460   246  
    1552   247  
    1548   248  
    1827   249  
    1737   250  
    1941   251  
    1474   252  
    1458   253  
    1542   254  
    1404   255  
    1522   256  
    1385   257  
    1641   258  
    1510   259  
    1681   260  
    1938   261  
    1868   262  
    1726   263  
    1456   264  
    1445   265  
    1456   266  
    1365   267  
    1487   268  
    1558   269  
    1488   270  
    1684   271  
    1594   272  
    1850   273  
    1998   274  
    2079   275  
    1494   276  
    1057   277  
    1218   278  
    1168   279  
    1236   280  
    1076   281  
    1174   282  
    1139   283  
    1427   284  
    1487   285  
    1483   286  
    1513   287  
    1357   288  
    1165   289  
    1282   290  
    1110   291  
    1297   292  
    1185   293  
    1222   294  
    1284   295  
    1444   296  
    1575   297  
    1737   298  
    1763   299  
end

tsset t, monthly
gen lgksi = log(ksi)

sspace (u L.u, state noerror noconstant) (lgksi u, noconstant)

To save on scrolling, this is the sspace command I'm using

sspace (u L.u, state noerror noconstant) (lgksi u, noconstant)

This gives me a r(498) error of "state equations must have at least one error term". Do I need to use a constraint here to estimate this version of the model? I know I *can* estimate it in a much easier way, but since this is a learning experience, I want to know if it's possible to estimate it using the sspace command.

In the second model, $\xi_t$ is allowed to vary, so we're simply estimating the basic local level model. This is my code:

clear

input t norway
       t   norway  
    1970      560  
    1971      533  
    1972      490  
    1973      511  
    1974      509  
    1975      539  
    1976      471  
    1977      442  
    1978      434  
    1979      437  
    1980      362  
    1981      338  
    1982      401  
    1983      409  
    1984      407  
    1985      402  
    1986      452  
    1987      398  
    1988      378  
    1989      381  
    1990      332  
    1991      323  
    1992      325  
    1993      281  
    1994      283  
    1995      305  
    1996      255  
    1997      303  
    1998      352  
    1999      304  
    2000      341  
    2001      275  
    2002      312  
    2003      280  
end

g lgnorway = log(norway)
tsset t, yearly

sspace (u L.u, state noconstant) (lgnorway u, noconstant)

but this code runs without convergence for hours if I don't stop it. It displays output like this:

Iteration 394: log likelihood =  17.749084  (not concave)
Iteration 395: log likelihood =  17.749084  (not concave)
Iteration 396: log likelihood =  17.749084  (not concave)
Iteration 397: log likelihood =  17.749084  (not concave)
Iteration 398: log likelihood =  17.749084  (not concave)
Iteration 399: log likelihood =  17.749084  (not concave)
Iteration 400: log likelihood =  17.749084  (not concave)
Iteration 401: log likelihood =  17.749084  (not concave)
Iteration 402: log likelihood =  17.749084  (not concave)
Iteration 403: log likelihood =  17.749084  (not concave)

Am I misunderstanding how to use the sspace syntax, or am I doing something else wrong?

Thank you,
Michael Anbar

P.S. As a side note, mathjax rendering doesn't seem to work in previews on Windows 7/Firefox 31.

[Michael Anbar]

Is there other information I can add to this post to make it more helpful? I read the FAQ, and as far as I can tell I've followed its guidelines, but I still haven't found a solution.

Thank you,
Michael Anbar

[Michael Anbar]

This was also posted here too: http://stats.stackexchange.com/q/113206/41138

[Clyde Schechter]

Disclaimer: I don't know anything about the -sspace- command. I didn't even know it exists until I saw your post.

But one general trick to help debug a non-converging estimation command in Stata is to add the -trace- option to the command, and also the -iterate(#)- option with # specified as some iteration number where you have reached the apparently endless loop, but, to get there quickly, not too far into that.

The command will run and you will get updated estimates of the model parameters after each iteration. Inspecting that may identify a parameter that has clearly gone into absurd territory, or one for which standard errors are not being estimated. That can help you identify something that might need to be omitted from the model, or transformed in some way, or requires specifying a starting value, or something like that.

I know that's pretty vague and general, but it's a technique that often gives great insight into a non-convergent estimation.

If that is no help, another technique that sometimes works, generically, for non-convergent estimations is to use the -difficult- option, or to choose a non-default value of the -method- option for the command.