I am learning about time series and forecasting and I stumbled across exponential smoothing and other derived methods. In an exponential smoothing model, each prediction is given by a level equation together with a smoothing equation, namely $$\begin{array}{l} \tilde{y}_{t+1} = \ell_t, \\ \ell_t = \alpha y_{t-1} + (1-\alpha)\ell_{t-1}. \end{array}$$ The forecaster can guess $\alpha$ based in different reasons and not all of them are necessarely purely mathematical, but it is usual to estimate it by minimizing the sum of the squared errors. That is, considering a one-step forecast model and then miniziming the sum $$\sum_{t = 2}^N e_t^2 = \sum_{t=2}^N\left(\tilde{y}_t-y_t\right)^2$$ where in this one-step model we consider $\ell_{t-2}=y_{t-2}$, so we have $$\tilde{y}_t=\ell_t=\alpha y_{t-1} + (1-\alpha)y_{t-2}.$$ Performing the necessary calculations we can derive the formula $$\alpha = \cfrac{ \sum_{t=2}^N(y_{t}-y_{t-2})(y_{t-1}-y_{t-2})}{ \sum_{t=2}^N (y_{t-1}-y_{t-2})^2}.$$ Exponential smoothing can be upgraded to a method that takes trends into account, this is the so-called Holt's linear trend method and is given by the following expressions: $$\begin{array}{l} \tilde{y}_{t+h} = \ell_t + hb_t, \\ \ell_t = \alpha y_{t-1} + (1-\alpha)(\ell_{t-2} + b_{t-2}),\\ b_t=\beta(\ell_t-\ell_{t-1})+(1-\beta)b_{t-2}. \end{array}$$ In order to estimate the parameters $\alpha$ and $\beta$, all the literature I've been able to read suggests to follow the same path as with regular exponential smoothing, that is, minimizing the squared error, but none does actually give any insights on the process. This is where I decided to try it myself, taking the same approach: we consider a one-step forecast model, so $h=1$ and the equations are thus given by $$\begin{array}{l} \tilde{y}_{t} = \ell_{t-1} + b_{t-1}, \\ \ell_{t-1} = \alpha y_{t-1} + (1-\alpha)(2y_{t-2} - y_{t-1}),\\ b_{t-1}=\beta(\ell_{t-1}-y_{t-2})+(1-\beta)(y_{t-2}-y_{t-1}). \end{array}$$ More precissely, we are considering $\ell_{t-2} = y_{t-2}$ and $b_{t-2} = y_{t-2}-y_{t-1}$ ($b_t$ is meant to be the slope of the straight line going through the last couple of points before the forecast one).

Minimizing the squred error now requires performing multi-variable optimization, so taking both derivatives and equalising to zero I get, after many calculations, the following formula: $$\beta = \frac{ \sum_{t=3}^N (2y_{t-2}-y_{t-3}-y_{t-1})(3y_{t-2}-2y_{t-3}-y_t)}{\alpha \sum_{t=3}^N (y_{t-1}-2y_{t-2}+y_{t-3})^2} - 1.$$ Now, the problem here is that I get the same expression for both derivatives, namely that both $$\frac{\partial}{\partial \alpha} \sum_{t=3}^N e_t^2 = 0$$ and $$\frac{\partial}{\partial \beta} \sum_{t=3}^N e_t^2 = 0$$ will lead to the same formula, implying that the system has infinitely many solutions and that there is actually a whole curve of minima.

Personally, I find this hard to believe but no matter how much I look to my computations I won't find any mistakes. My question is if my reasoning is correct and if somebody could verify my approach. In case that this is indeed correct, would it be reasonable to estimate $\alpha$ as if in regular exponential smoothing and then use this parameter to estimate a $\beta$ for Holt's method?

Thank you very much.


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