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, 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?

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    $\begingroup$ 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$. $\endgroup$ – Andrew M Oct 19 '19 at 21:28
  • $\begingroup$ Also $\gamma_0$ is not identifiable separately from $\beta_0$ so that might be part of your problem in terms of interpreting the coefficients. $\endgroup$ – Andrew M Oct 19 '19 at 21:39
  • $\begingroup$ @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. $\endgroup$ – mark leeds Oct 20 '19 at 20:30
  • $\begingroup$ @AndrewM Yep - I've edited the typo. Re identification, there is plenty of empirical work which does so, for instance: bankofengland.co.uk/-/media/boe/files/working-paper/2003/… $\endgroup$ – Thev Oct 21 '19 at 0:02
  • $\begingroup$ @TheveshTheva : The coefficient multiplying the $\theta$ still isn't time aligned in that $C_t$ is one step behind $Y_t$. $\endgroup$ – mark leeds Oct 21 '19 at 1:28

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