这是一份nottingham诺丁汉大学PSGY1004作业代写的成功案例

The probability of an event for the ith individual can be written as a joint probability of $y_{i t}$.
$$
\operatorname{Pr}\left(y_{i 1}, \ldots, y_{i T}\right)=\int_{a_{i 1}}^{b_{i 1}} \ldots \int_{a_{i T}}^{b_{i T}} \phi\left(\varepsilon_{i 1}, \ldots, \varepsilon_{i T}\right) d \varepsilon_{i T} \ldots d \varepsilon_{i 1}
$$
where $a_{i t}=-\mathbf{x}{i t}^{\prime} \boldsymbol{\beta}, b{i t}=+\infty$ if $y_{i t}=1$, and $a_{i t}=-\infty, b_{i t}=-\mathbf{x}{i t}^{\prime} \beta$ if $y{i t}=0$, and $\phi(\cdot)$ is the standard $T$-variate normal density function. Conditioning on the permanent component, $v_{i}$, this expression can be written as
$$
\int_{a_{i 1}}^{b_{i 1}} \ldots \int_{a_{i T}}^{b_{i T}} \int_{-\infty}^{+\infty} \phi\left(u_{i 1}, \ldots, u_{i T} \mid v_{i}\right) \phi\left(v_{i}\right) d v_{i} d u_{i 1} \ldots d u_{i T}
$$
Because the transitory components are independent conditional on $v_{i}$, this expression can be simplified. In terms of model quantities, can be written as
$$
\operatorname{Pr}\left(y_{i 1}, \ldots, y_{i T}\right)=\int_{-\infty}^{+\infty} \prod_{t=1}^{T} \Phi\left(\mathbf{x}{i t}^{\prime} \beta \mid v{i}\right)^{y_{i t}} \Phi\left(-\mathbf{x}{i t}^{\prime} \beta \mid v{i}\right)^{1-y_{i t}} \phi\left(v_{i}\right) d v_{i},
$$

PSGY1004 COURSE NOTES ：

Since we estimate $J-1$ coefficients for any explanatory variable we still have not utilized the ordinal information in the dependent variable. If we further assume that the dependent variable can be scaled into an interval variable, $y_{j}$, model $6.8$ can be further simplified to
$$
\log \left(\frac{p_{j}}{p_{j-1}}\right)=\alpha_{j}+\beta\left(y_{j}-y_{j-1}\right) x_{i}, \quad i=1, \ldots, I, \quad j=2, \ldots, J
$$
As a special case, if integer scoring is applied to both $y$ and $x$, is reduced to
$$
\log \left(\frac{p_{j}}{p_{j-1}}\right)=\alpha_{j}+\beta i, \quad i=1, \ldots, I ; \quad j=2, \ldots, J
$$