Also testing equations

[Craig Kerr]

written without a wrapper: \[ \frac{2}{3}+ \mathbf{\pi} \] Using code wrapper:

   
  \[ \frac{2}{3}+ \mathbf{\pi} \]

[Craig Kerr]

I am attempting to use Dixon & Coles (1997) method of predicting soccer matches but rather than estimate a model predicting both team's goals scored (bivariate poisson) I'm simply predicting the game outcome (home team win, lose, or draw) using an ordered probit. The authors estimate a dynamic model where games in the past are down-weighted by a negative exponential function so that the psuedo-likliehood function in my research should be

\[ L=\prod_{j\in A_{t}}(\sum_{i=1}^{3}I_i(y_{j})p_{ij}(x_{j}))^{exp (-\xi t)} \]

for all games j that occurred before time t and outcome i where

\[I_i(y_j)= \left\{ \begin{array}{ll} 1& \textrm{if } y_j=i\\ \\ 0&\textrm{ otherwise } \end{array} \right. \]

and \[\xi\] is chosen to be a constant by the econometrician.

This yields a log-likelihood function of

\[\ln L=\sum_{j\in A_{t}}exp(-\xi t)\ln(\sum_{i=1}^{3}I_i(y_{j})p_{ij}(x_j))\]

which presumably can be estimated via maximum likelihood.

My question is, can I use iweights to accomplish this in Stata? My code is

oprobit y x [iw=xi]

Does this estimate the equation above as written and what, if anything, does iweights do to the estimated variance-covariance matrix?

According to the Methods and Formulas section from ologit (where readers are redirected for oprobit), optional weights enter the log-likelihood function as
\[ \ln L = \sum_{j=1}^Nw_{j}\sum_{i=1}^{k}I_i(y_j)\ln p_{ij} \] where \[ln p_{ij}\] is the natural log of the probability that observation i has outcome j, \[w_{j}\] are the optional weights so my guess is it is working as planned but I'm not 100% sure and there's no mention of what iweights do to the variance-covariance matrix.

[Craig Kerr]

I am attempting to use Dixon & Coles (1997) method of predicting soccer matches but rather than estimate a model predicting both team's goals scored (bivariate poisson) I'm simply predicting the game outcome (home team win, lose, or draw) using an ordered probit. The authors estimate a dynamic model where games in the past are down-weighted by a negative exponential function so that the psuedo-likliehood function in my research should be

\[ L=\prod_{j\in A_{t}}(\sum_{i=1}^{3}I_i(y_{j})p_{ij}(x_{j}))^{exp (-\xi (t-t_j))} \]

for all games j that occurred before time t and outcome i where

\[p_{ij}(x)\]

is the probability that game j has outcome i given game variables x,

\[I_i(y_j)= \left\{ \begin{array}{ll} 1& \textrm{if } y_j=i\\ \\ 0&\textrm{ otherwise } \end{array} \right. \]

and \[\xi\] is chosen to be a constant by the econometrician.

This yields a log-likelihood function of

\[\ln L=\sum_{j\in A_{t}}exp(-\xi (t-t_j))\ln(\sum_{i=1}^{3}I_i(y_{j})p_{ij}(x_j))\]

which presumably can be estimated via maximum likelihood.

My question is, can I use iweights to accomplish this in Stata? My code is

oprobit y $x [iw=xi]

where $x is the global for all game-level variables and xi is the variable I made to represent

\[exp(-\xi (t-t_j)) \]

for each game. Does this estimate the equation above as written and what, if anything, does the use of iweights do to the estimated variance-covariance matrix?

According to the Methods and Formulas section from ologit (where readers are redirected for oprobit), optional weights enter the log-likelihood function as

\[ \ln L = \sum_{j=1}^Nw_{j}\sum_{i=1}^{k}I_i(y_j)\ln p_{ij} \]

so my guess is it is working as planned but I'm not 100% sure and there's no mention of what iweights do to the variance-covariance matrix.