<p>For question 1), take the expectation of both sides, assume that the expectation of Y is the same in both periods (the definition of stationarity) and see what conditions the coefficients on X must satisfy.</p>

<p>For question 2), try rearranging the equation to be in terms of X and see what determines its variation.</p>

<p>P.S. I am still getting the hang of the mathematical notation on here, but I figure too much working out would be giving it away anyway</p>

CC BY-SA 3.0
2015-11-11T17:09:45.180Z
Thanks for your reply. Here is my attempt at question 1) $y_t = \alpha y_{t-1} + \beta_0 x_t + \beta_1 x_{t-1} + \epsilon_t$ Taking expectations  $E(y_t) = \alpha E(y_{t-1}) + \beta_0 E(x_t) + \beta_1 E(x_{t-1})$,  expectation of error term is zero. Then assuming stationarity, $E(y_t) = E(y_{t-1})$ which using the above equation happens when $\alpha =1 , \beta_0 =0$ and  $\beta_1 = 0$
damson_jam
CC BY-SA 3.0
2015-11-11T17:15:51.713Z
So $y_t$ could be stationary if these conditions are satisfied.
damson_jam
CC BY-SA 3.0
2015-11-11T17:27:27.120Z
For part 2) It first asks to write down the error correction model and show the relationship between the new parameters and those in the original model: $y_t = \alpha y_{t-1} + \beta_0 x_t + \beta_1 x_{t-1} + \epsilon_t $ subtract $y_{t-1}$ from both sides. $\Delta y_t = - (1- \alpha) y_{t-1} + \beta_0 x_t + \beta_1 x_{t-1} + \epsilon_t $ subtract and then add back $\beta_0 x_{t-1}$ from the RHS and re-write as $\Delta  y_t = \Delta \beta_0 x_t + \lambda(y_{t-1} - \delta x_{t-1})  + v_t $ where $\lambda = 1-\alpha$ and $\delta = \frac {\beta_0 + \beta_1}{1-\alpha}$
damson_jam
CC BY-SA 3.0
2015-11-11T17:39:13.243Z
For the next part of 2), when you say rearrange in terms of X, do you mean a)  $\beta_0 x_t + \beta_1 x_{t-1} = y_t -\alpha y_{t-1} - \epsilon_t $ or b)  $x_t = \frac {1}{\beta_0} (y_t -\alpha y_{t-1} - \beta_1 x_{t-1} - \epsilon_t) $ ? In either case im still having trouble progressing.
damson_jam
CC BY-SA 3.0
2015-11-13T14:05:19.390Z
Any more ideas for part 2?
damson_jam
CC BY-SA 3.0
2015-11-11T16:46:49.233Z
2015-11-11T16:46:49.233Z
globule
<p>This is a homework question: </p>

<p>$y_t = \alpha y_{t-1} + \beta_0 x_t + \beta_1 x_{t-1} + \epsilon_t , 0&lt;\alpha&lt;1, (t=1,...,T)$
where $x_t$ is an I(1) process independent from $\epsilon_t$ for all $s$ and $t$, and $\epsilon_t , t=1,...,T $ are i.i.d with mean zero and variance $\sigma^2$</p>

<p>Question</p>

<p>1) Suppose that $\beta_0 \not= -\beta_1$. Is it possible that $y_t$ is stationary?</p>

<p>2) Write down the error correction form of the model, showing the relationship between the new parameters and those in the original model. 
Argue that $y_t$ and $x_t$ are co-integrated and write down the co-integrating vector</p>

<p>Any help/pointers would be great.</p>

<econometrics>
Co-integration and Error correction
damson_jam
# Co-integration and Error correction

(Asked by: username_0 on 2015-11-11)

This is a homework question:

$y\_t = \alpha y\_{t-1} + \beta\_0 x\_t + \beta\_1 x\_{t-1} + \epsilon\_t , 0<\alpha<1, (t=1,...,T)$
where $x\_t$ is an I(1) process independent from $\epsilon\_t$ for all $s$ and $t$, and $\epsilon\_t , t=1,...,T $ are i.i.d with mean zero and variance $\sigma^2$

Question

1) Suppose that $\beta\_0 \not= -\beta\_1$. Is it possible that $y\_t$ is stationary?

2) Write down the error correction form of the model, showing the relationship between the new parameters and those in the original model.
Argue that $y\_t$ and $x\_t$ are co-integrated and write down the co-integrating vector

Any help/pointers would be great.

**Tags:** econometrics

---

## 1 answers

### Answer by username_1 on 2015-11-11. Score: 1

For question 1), take the expectation of both sides, assume that the expectation of Y is the same in both periods (the definition of stationarity) and see what conditions the coefficients on X must satisfy.

For question 2), try rearranging the equation to be in terms of X and see what determines its variation.

P.S. I am still getting the hang of the mathematical notation on here, but I figure too much working out would be giving it away anyway

> **Comments:**
> **username_0**: Thanks for your reply. Here is my attempt at question 1) $y_t = \alpha y_{t-1} + \beta_0 x_t + \beta_1 x_{t-1} + \epsilon_t$ Taking expectations  $E(y_t) = \alpha E(y_{t-1}) + \beta_0 E(x_t) + \beta_1 E(x_{t-1})$,  expectation of error term is zero. Then assuming stationarity, $E(y_t) = E(y_{t-1})$ which using the above equation happens when $\alpha =1 , \beta_0 =0$ and  $\beta_1 = 0$  
> **username_0**: So $y_t$ could be stationary if these conditions are satisfied.  
> **username_0**: For part 2) It first asks to write down the error correction model and show the relationship between the new parameters and those in the original model: $y_t = \alpha y_{t-1} + \beta_0 x_t + \beta_1 x_{t-1} + \epsilon_t $ subtract $y_{t-1}$ from both sides. $\Delta y_t = - (1- \alpha) y_{t-1} + \beta_0 x_t + \beta_1 x_{t-1} + \epsilon_t $ subtract and then add back $\beta_0 x_{t-1}$ from the RHS and re-write as $\Delta  y_t = \Delta \beta_0 x_t + \lambda(y_{t-1} - \delta x_{t-1})  + v_t $ where $\lambda = 1-\alpha$ and $\delta = \frac {\beta_0 + \beta_1}{1-\alpha}$  
> **username_0**: For the next part of 2), when you say rearrange in terms of X, do you mean a)  $\beta_0 x_t + \beta_1 x_{t-1} = y_t -\alpha y_{t-1} - \epsilon_t $ or b)  $x_t = \frac {1}{\beta_0} (y_t -\alpha y_{t-1} - \beta_1 x_{t-1} - \epsilon_t) $ ? In either case im still having trouble progressing.  
> **username_0**: Any more ideas for part 2?  

---

