# How to get a forecast equation for $\hat{y}$ using ETS state space model

The ets(AAA) state space model (Rob Hyndman's handbook) is as below

State equation is $$$$Y_t = L_{t-1} + b_{t-1} + S_{t - m} + \varepsilon_t$$$$

The measurement equations are $$$$L_t = L_{t-1} + b_{t-1} + \alpha\varepsilon_t$$$$

$$$$b_t = b_{t-1} + \beta\varepsilon_t$$$$

$$$$S_t = S_{t-m} + \gamma\varepsilon_t$$$$

My question is, how do I get 1 step ahead forecast from these equations considering that error component is not known for $${\hat{Y}_{t+1}}$$.

• Hi: This is also called the single source of error state space model. ( a kalman filter ). In order to get forecasts, you first need to estimate the filter given your data. I don't know if Rob wrote a package for doing this ? If he didn't, then you can use the DLM package inR but it will be tricky to construct the matrices correctly. I can't go into details here. Read either Rob's book or the paper at the link below. monash.edu/business/econometrics-and-business-statistics/…. – mlofton Apr 14 at 17:07
• Thanks Mlofton. I am aware of DLM package. My question was actually very much to that line.... If I am correct, I can use this as a state space model to calculate the result just like the case given below? stats.stackexchange.com/questions/212129/… – Biswajit Jana Apr 14 at 18:24
• I have got my answer in this book - Forecasting with Exponential Smoothing The State Space Approach Authors: Hyndman, R., Koehler, A.B., Ord, J.K., Snyder, R.D. – Biswajit Jana Apr 15 at 6:22
• Hi: The link you pointed to is exactly what you don't want to use because that the standard KF where there are two sources of noise: one for the observation and one for system. the ETS model which is also known as single source of error state space IS A DIFFERENT formulation. For certain models ( arima(0,1,1) and arima(0,2,2) which are random walk plus noise and local linear trend, they can result in differnt answers. I can't get into this in a comment but the estimation details are different if one uses the DLM package. – mlofton Apr 15 at 16:12
• Hi: I have that book but I don't know what approach Rob uses. Assuming that he uses an exponential smoothing approach to estimate it, then that that's fine and more straightforward. Using the DLM package will be trickier ( you can't use the C and W that giovanni uses in the book because that's for the model at the link which is different ) but also possible. – mlofton Apr 15 at 16:17