<p>Say that we are regressing consumption <span class="math-container">$C_t$</span> on time <span class="math-container">$Y_t$</span>. Furthermore, suppose that both series are <span class="math-container">$I(1)$</span> and are co-integrated.</p>

<p>Given this, we set up the error correction model (ECM) as follows: <span class="math-container">$$\Delta C_t = \beta_0 + \beta_1 \Delta Y_t + \theta(C_{t-1} - \gamma_0 - \gamma_1 Y_t) + u_t$$</span> which is estimated using the 2-step Engle-Granger approach.</p>

<p>My question is: What is the economic interpretation of <span class="math-container">$\gamma_1$</span>? Surely we cannot interpret this as the marginal propensity to consume (MPC)? I interpret it as describing the long-run equilibrium of the system.</p>

<p>However, I'm struggling to explain why this is the case, since mathematically: <span class="math-container">$$\gamma_1 = \frac{\partial C_t}{\partial Y_t}$$</span> which is exactly the definition of the MPC.</p>

<hr>

<p>On <a href="https://en.wikipedia.org/wiki/Error_correction_model#An_example_of_ECM" rel="nofollow noreferrer">the Wikipedia page on ECMs</a>, they describe <span class="math-container">$\gamma_1$</span> as the average propensity to consume (APC), which makes a lot more sense to, but which doesn't sit well with me since mathematically: <span class="math-container">$$\text{APC} = \frac{C}{Y}$$</span> and clearly that is not the definition of <span class="math-container">$\gamma_1$</span> since: <span class="math-container">$$\frac{C}{Y} = \frac{\gamma_0}{Y} + \gamma1$$</span></p>

<p>Could someone help me sort out my confusion?</p>

CC BY-SA 4.0
2019-10-19T21:28:29.050Z
I think there is an error in your formula. Shouldn't it be $\Delta Y_t$  rather than $Y_t$ outside of the long-run relationship and inside it should be $Y_{t-1}$. As it currently stands your long-run coefficient of interest is unidentified and the equation is econometrically unbalanced since $\Delta C_t$ is of lower order integration than $Y_t$ which will force $ \beta_1 =0$.
Andrew M
CC BY-SA 4.0
2019-10-19T21:39:46.670Z
Also $\gamma_0$ is not identifiable separately from $\beta_0$ so that might be part of your problem in terms of interpreting the coefficients.
Andrew M
CC BY-SA 4.0
2019-10-20T20:30:57.357Z
@Andrew M: I think the OP might be using $Y_t$ for what is usually ( in the literature ) $X_t$ and $C_t$ for what is usually $Y_t$. But you might be correct. I'm not sure because I'm not used to that notation.
mark leeds
CC BY-SA 4.0
2019-10-21T00:02:12.297Z
@AndrewM Yep - I've edited the typo. Re identification, there is plenty of empirical work which does so, for instance: https://www.bankofengland.co.uk/-/media/boe/files/working-paper/2003/the-dynamics-of-consumers-expenditure-the-uk-consumption-ecm-redux
Thev
CC BY-SA 4.0
2019-10-21T01:28:21.457Z
@TheveshTheva : The coefficient multiplying the $\theta$ still isn't time aligned in that $C_t$ is one step behind  $Y_t$.
mark leeds
CC BY-SA 4.0
2019-10-22T00:31:15.583Z
@TheveshTheva if you start from the model $C_t$ = $\beta_0$ + $\beta_1$ $Y_t$ + $\alpha$ $C_{t-1}$ then the equivalent ECM representation is $\Delta$ $C_t$ =  $\beta_0$ + $\beta_1$ $\Delta Y_t$ + ($\alpha$-1)($C_{t-1}$- $\frac{\beta_1}{1-\alpha} Y_{t-1}$) which implies that $\gamma_0=0$ . Alternatively it can written as  $\Delta$ $C_t$ = $\beta_1$ $\Delta Y_t$ + ($\alpha$-1)($C_{t-1}$- $\frac{\beta_0}{1-\alpha}$ -$\frac{\beta_1}{1-\alpha} Y_{t-1}$ ) which implies that $\gamma_0\neq0$. Both representations are equivalent.
Andrew M
<econometrics><applied-econometrics><consumption>
Interpretation of coefficients in ECM for consumption
Thev
# Interpretation of coefficients in ECM for consumption

(Asked by: username_0 on 2019-10-19)

Say that we are regressing consumption $C\_t$ on time $Y\_t$. Furthermore, suppose that both series are $I(1)$ and are co-integrated.

Given this, we set up the error correction model (ECM) as follows: $$\Delta C\_t = \beta\_0 + \beta\_1 \Delta Y\_t + \theta(C\_{t-1} - \gamma\_0 - \gamma\_1 Y\_t) + u\_t$$ which is estimated using the 2-step Engle-Granger approach.

My question is: What is the economic interpretation of $\gamma\_1$? Surely we cannot interpret this as the marginal propensity to consume (MPC)? I interpret it as describing the long-run equilibrium of the system.

However, I'm struggling to explain why this is the case, since mathematically: $$\gamma\_1 = \frac{\partial C\_t}{\partial Y\_t}$$ which is exactly the definition of the MPC.

---

On [the Wikipedia page on ECMs](https://en.wikipedia.org/wiki/Error_correction_model#An_example_of_ECM), they describe $\gamma\_1$ as the average propensity to consume (APC), which makes a lot more sense to, but which doesn't sit well with me since mathematically: $$\text{APC} = \frac{C}{Y}$$ and clearly that is not the definition of $\gamma\_1$ since: $$\frac{C}{Y} = \frac{\gamma\_0}{Y} + \gamma1$$

Could someone help me sort out my confusion?

**Tags:** econometrics, applied-econometrics, consumption

> **Comments:**
> **username_1**: I think there is an error in your formula. Shouldn't it be $\Delta Y_t$  rather than $Y_t$ outside of the long-run relationship and inside it should be $Y_{t-1}$. As it currently stands your long-run coefficient of interest is unidentified and the equation is econometrically unbalanced since $\Delta C_t$ is of lower order integration than $Y_t$ which will force $ \beta_1 =0$.  
> **username_1**: Also $\gamma_0$ is not identifiable separately from $\beta_0$ so that might be part of your problem in terms of interpreting the coefficients.  
> **username_2**: @username_1: I think the OP might be using $Y_t$ for what is usually ( in the literature ) $X_t$ and $C_t$ for what is usually $Y_t$. But you might be correct. I'm not sure because I'm not used to that notation.  
> **username_0**: @username_1 Yep - I've edited the typo. Re identification, there is plenty of empirical work which does so, for instance: https://www.bankofengland.co.uk/-/media/boe/files/working-paper/2003/the-dynamics-of-consumers-expenditure-the-uk-consumption-ecm-redux  
> **username_2**: @TheveshTheva : The coefficient multiplying the $\theta$ still isn't time aligned in that $C_t$ is one step behind  $Y_t$.  
> **username_1**: @TheveshTheva if you start from the model $C_t$ = $\beta_0$ + $\beta_1$ $Y_t$ + $\alpha$ $C_{t-1}$ then the equivalent ECM representation is $\Delta$ $C_t$ =  $\beta_0$ + $\beta_1$ $\Delta Y_t$ + ($\alpha$-1)($C_{t-1}$- $\frac{\beta_1}{1-\alpha} Y_{t-1}$) which implies that $\gamma_0=0$ . Alternatively it can written as  $\Delta$ $C_t$ = $\beta_1$ $\Delta Y_t$ + ($\alpha$-1)($C_{t-1}$- $\frac{\beta_0}{1-\alpha}$ -$\frac{\beta_1}{1-\alpha} Y_{t-1}$ ) which implies that $\gamma_0\neq0$. Both representations are equivalent.  

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## 0 answers

