# Level and trend in Holt's linear trend method

I'm studying about exponential smoothing methods and something I still can not understand is the behavior of level and trend components when you increase and decrease level and trend smoothing parameters.

Let $\alpha_1$ and $\alpha_2$ be the smoothing parameter for the level and $\beta_1$ and $\beta_2$ the smoothing parameter for the trend. Given the following two examples, where the first is with $\alpha_1$ and $\beta_1$ and the second with $\alpha_2$ and $\beta_2$.

Is $\alpha_1>\alpha_2$ and $\beta_1>\beta_2$? If yes why and if not why?

I was testing various combinations of coefficients to try to identify a behavior in the components, but I could not come to some conclusion yet. I know that assigning larger values to the parameters means assigning more weight to the more recent observations, but it seems to me that at some point some distortion of the components may occur.

You are right that increasing a smoothing parameter will give a higher weight to more recent observations. Here is an alternative way of putting this: with a higher weight, the corresponding (level or season) component will change more quickly. It will be more volatile. You can see this most easily in single exponential smoothing, using the weighted average form:

$$\ell_t=\alpha y_t+(1-\alpha)\ell_{t-1}.$$

Higher $\alpha$s give more weight to the last observation and less to the previous level component. At the extreme case of $\alpha=0$, the level component will never change, and at the other extreme of $\alpha=1$, the level component changes maximally at each time point, namely to the last observation.

The same relationship between smoothing parameters and component volatility holds for more complex exponential smoothing schemes.

Now, let's look at your plots. The middle panel in each case shows the level component over time. The level component in the bottom plot looks more volatile (e.g., in 1977-1980, several "kinks" appear in the bottom level component that do not appear in the top one). So it looks like the bottom plot probably came from a higher level smoothing constant, $\alpha_2>\alpha_1$.

For the trend ("slope") component, the bottom plot again looks a little more volatile, but here it's less obvious. I would still assume that $\beta_2>\beta_1$.

• I think I get the idea, but it's pretty hard to see these behaviors graphically. In this case $\alpha_1=0.4$, $\beta_1=0.9$, $\alpha_2=0.8$ and $\beta_2=0.4$ – user72621 Nov 18 '17 at 19:52
• Interesting. There may be some interaction going on for the $\beta$ part. – Stephan Kolassa Nov 18 '17 at 19:54