How do I prove that ARIMA(0,1,1) is equivalent to simple exponential smoothing?

Question

this is an exam question of mine, but I am really struggling with it. I have seen proofs online but they are too vague and do not connect the dots explicitly. Would someone be able to post a proof which shows the steps exactly?

Answer 1

An Arima(0,1,1) can be written as $$ X_t = X_{t-1} + \epsilon_t + \theta\epsilon_{t-1}. $$ I think if you use the innovations algorithm to come up with minimum-MSE, recursive, one-step-ahead forecasts, you will get $$ \hat{X}_t = X_{t-1} + \theta (X_{t-1} - \hat{X}_{t-1}) $$ for $t > 1$. If you just define $ \alpha - 1 = \theta$ then this equation above can be rewritten as $$ \hat{X}_t = \alpha X_{t-1} + (1-\alpha)\hat{X}_{t-1}. $$

Answer 2

Apologies if this answer seems a bit jumbled up.

It is likely not the simplest way of doing it, but it is complete and from basics, so should hopefully make sense.

Apologies for the formatting - I'm new to this (underscores are meant to signify that whatever is written next is subscript, i.e. e_(t-1) is e subscript "t-1")

I was actually looking for an answer to this myself and found a form of explanation here: https://people.duke.edu/~rnau/411arim.htm

In effect:

ARIMA(0,1,1) has the general form: (1-B) Y_t = θ_0 +(1 - θ_1 B) e_t

Where:

Y_t is data value at t

e_t is error at t

θ_0 and θ_1 are constants

B is the backshift operator [converts a value to one period back - i.e. B Y_t =Y_(t-1)] (If you don’t understand that you may recognise the formula below)

This can be expanded out to the following:

Y_t -BY_t = θ_0 + e_t - θ_1 B e_t

Y_t - Y_(t-1) = θ_0 + e_t - θ_1 e_(t-1)

Y_t = Y_(t-1 )+ θ_0 + e_t - θ_1 e_(t-1)

For forecasting the formula would then become:

F_t = Y_t + θ_0 - θ_1 e_t

(All the non-constant terms are increased by one period. The error term e_t becomes e_(t+1), but as you cannot forecast a future error this is removed)

That is the generalised forecasting formula for ARIMA(0,1,1). You may be given/know this already. Obviously, I cannot say for sure, but in an exam you may not have to go through that whole derivation.

Now looking on the SES side:

The generalised SES formula for forecasting is:

F_(t+1) = α Y_t + (1-α) F_t

This can be rearranged as follows:

F_(t+1) = α Y_t + F_t - α F_t

F_(t+1)= α (Y_t - F_t) + F_t

We then state that:

α = 1 - θ_1

This means that:

F_(t+1) = (1 - θ_1) (Y_t - F_t) + F_t

F_(t+1) = Y_t - F_t - θ_1 Y_t + θ_1 F_t + F_t

F_(t+1) = Y_t - θ_1 Y_t + θ_1 F_t

F_(t+1) = Y_t - θ_1 (Y_t - F_t)

We know that (Y_t - F_t) = e_t

(This is just the definition of an error)

Therefore:

F_(t+1) = Y_t - θ_1 e_t

This formula is the same as the generalised ARIMA(0,1,1) apart from the θ_0 term. This is a constant though, and a constant can be zero. Therefore, SES can be said to be equivalent to an ARIMA(0,1,1) model without a constant (i.e. θ_0 = 0), where α = 1 - θ_1.

Hope this helps!