<p>The derivation of the score process is correct. To verify that the process is a Martingale, recall the <a href="https://en.wikipedia.org/wiki/Martingale_(probability_theory)#Definitions" rel="nofollow">definition</a>. It becomes clear that if we substitute $W_{t+1}$ back into the equation
$$
s_t(\theta \mid \textbf X) =
\begin{bmatrix}
 (1 - \beta_0) \sum_{j=1}^t W_j \\
 \sum_{j=1}^t W_j (X_{j-1} - \alpha_0)
\end{bmatrix}.
$$
Because $W_{t+1}$ are Normal with mean 0 and variance 1 (I'm assuming they're iid), then
$$
E[s_{t+1} \mid s_t ] = s_t + E \left [
\begin{matrix}
(1 - \beta_0) W_{t+1} \\
W_{t+1} (X_t - \alpha_0)
\end{matrix} \middle |
s_t
\right ]
= s_t
$$ 
and we are finished.</p>

CC BY-SA 3.0
2015-01-05T19:29:52.803Z
2015-09-25T20:23:50.240Z
2015-09-25T20:23:50.240Z
jmbejara
<p>The following is a specific question that is useful for demonstrating a general idea.</p>

<p>Consider the following autoregressive model:
$$
X_{t+1} = \alpha_0 + \beta_0 (X_t - \alpha_0) + W_{t+1},
$$
where $-1 &lt; \beta_0 &lt; 1$ and $W_{t+1}$ is distributed as a normal with mean zero and variance one. </p>

<p>How should I go about constructing the bivariate score process associated with the parameters $\alpha_0$ and $\beta_0$? How can I verify that it is a martingale?</p>

<p><strong>Progress:</strong></p>

<p>I would begin by constructing the log-likelihood process as follows (conditioning on $X_0$):
$$
\ell_t(\theta \mid \textbf X) = -\frac t2 \ln(2 \pi) - \frac 12 \sum_{j=1}^t
   (X_j - \alpha_0 - \beta_0(X_{j-1} - \alpha_0))^2.
$$
Then, the score process can be constructed as
$$
s_t(\theta \mid \textbf X) =
\begin{bmatrix}
 (1 - \beta_0) \sum_{j=1}^t (X_j - \alpha_0 - \beta_0(X_{j-1} - \alpha_0)) \\
 \sum_{j=1}^t (X_j - \alpha_0 - \beta_0(X_{j-1} - \alpha_0)) (X_{j-1} - \alpha_0)
\end{bmatrix}.
$$
Is this correct? How do I proceed?</p>

<econometrics><time-series><self-study><martingales>
How do I construct the score process of a Markov model and verify that it is a Martingale?
jmbejara
# How do I construct the score process of a Markov model and verify that it is a Martingale?

(Asked by: username_0 on 2014-12-31)

The following is a specific question that is useful for demonstrating a general idea.

Consider the following autoregressive model:
$$
X\_{t+1} = \alpha\_0 + \beta\_0 (X\_t - \alpha\_0) + W\_{t+1},
$$
where $-1 < \beta\_0 < 1$ and $W\_{t+1}$ is distributed as a normal with mean zero and variance one.

How should I go about constructing the bivariate score process associated with the parameters $\alpha\_0$ and $\beta\_0$? How can I verify that it is a martingale?

**Progress:**

I would begin by constructing the log-likelihood process as follows (conditioning on $X\_0$):
$$
\ell\_t(\theta \mid \textbf X) = -\frac t2 \ln(2 \pi) - \frac 12 \sum\_{j=1}^t
(X\_j - \alpha\_0 - \beta\_0(X\_{j-1} - \alpha\_0))^2.
$$
Then, the score process can be constructed as
$$
s\_t(\theta \mid \textbf X) =
\begin{bmatrix}
(1 - \beta\_0) \sum\_{j=1}^t (X\_j - \alpha\_0 - \beta\_0(X\_{j-1} - \alpha\_0)) \\
\sum\_{j=1}^t (X\_j - \alpha\_0 - \beta\_0(X\_{j-1} - \alpha\_0)) (X\_{j-1} - \alpha\_0)
\end{bmatrix}.
$$
Is this correct? How do I proceed?

**Tags:** econometrics, time-series, self-study, martingales

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## 1 answers

### Answer by username_0 on 2015-01-05. Score: 4 | Accepted answer

The derivation of the score process is correct. To verify that the process is a Martingale, recall the [definition](https://en.wikipedia.org/wiki/Martingale_(probability_theory)#Definitions). It becomes clear that if we substitute $W\_{t+1}$ back into the equation
$$
s\_t(\theta \mid \textbf X) =
\begin{bmatrix}
(1 - \beta\_0) \sum\_{j=1}^t W\_j \\
\sum\_{j=1}^t W\_j (X\_{j-1} - \alpha\_0)
\end{bmatrix}.
$$
Because $W\_{t+1}$ are Normal with mean 0 and variance 1 (I'm assuming they're iid), then
$$
E[s\_{t+1} \mid s\_t ] = s\_t + E \left [
\begin{matrix}
(1 - \beta\_0) W\_{t+1} \\
W\_{t+1} (X\_t - \alpha\_0)
\end{matrix} \middle |
s\_t
\right ]
= s\_t
$$
and we are finished.


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