I received this email today:

I have a ques­tion about the ets() func­tion in R, which I am try­ing to use for Holt-​​Winters expo­nen­tial smooth­ing.
My prob­lem is that I am get­ting very dif­fer­ent esti­mates of the alpha, beta and gamma para­me­ters using ets() com­pared to HoltWin­ters(), and I can’t fig­ure out why.

This is a com­mon ques­tion, so I thought the answer might be of suf­fi­cient inter­est to post here.

There are sev­eral issues involved.

1. HoltWinters() and ets() are opti­miz­ing dif­fer­ent cri­te­rion. HoltWinters() is using heuris­tic val­ues for the ini­tial states and then esti­mat­ing the smooth­ing para­me­ters by opti­miz­ing the MSE. ets() is esti­mat­ing both the ini­tial states and smooth­ing para­me­ters by opti­miz­ing the like­li­hood func­tion (which is only equiv­a­lent to opti­miz­ing the MSE for the lin­ear addi­tive models).
2. The two func­tions use dif­fer­ent opti­miza­tion rou­tines and dif­fer­ent start­ing val­ues. That wouldn’t mat­ter if the sur­faces being opti­mized were smooth, but they are not. Because the MSE and like­li­hood sur­faces are both fairly bumpy, it is easy to find a local opti­mum. The only way to avoid this prob­lem is to use a much slower com­pu­ta­tional method such as PSO.
3. ets() searches over a restricted para­me­ter space to ensure the result­ing model is fore­castable. HoltWinters() ignores this issue (it was writ­ten before the prob­lem was even dis­cov­ered). See this paper for details (equiv­a­lently chap­ter 10 of my expo­nen­tial smooth­ing book).

I have exper­i­mented with many dif­fer­ent choices of the start­ing val­ues for the ini­tial val­ues and smooth­ing para­me­ters, and what is imple­mented in ets() seems about as good as is pos­si­ble with­out using a much slower opti­miza­tion rou­tine. Where there is a dif­fer­ence between ets() and HoltWinters(), the results from ets() are usu­ally more reliable.

A related ques­tion on esti­ma­tion of ARIMA mod­els was dis­cussed at http://​rob​jhyn​d​man​.com/​r​e​s​e​a​r​c​h​t​i​p​s​/​e​s​t​i​m​a​tion/.

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• Sta­tis­ti­cal tests for vari­able selection