# Objective function in linked prediction models

I would like to use R “optimize” function to estimate the model:

$$\hat{Y_{t}}=a\cdot \hat{Y}_{t-1}+b\cdot X_{t}+c\cdot X_{t-1}$$

Important thing is that the predition $$\hat{Y_{t}}$$ (let’s say prediction of Y on Tuesday) is calculated on the basis of former prediction $$\hat{Y}_{t-1}$$ (not real value of $$Y_{t-1}$$) in time t-1 (Monday).

What is key word to google to find out more on this class of models? I mean time series models where prediction is based on former prediction, not actual value like in ARIMA models.

With an objective function I would like to minimize the sum of the squares of the errors between $${Y_{t}}$$ and $$\hat{Y}_{t}$$. The objective function in one equation would look like endless spaghetti.

How to write objective function in R?

• This sounds like an ARIMA model. – Zach Mar 1 '12 at 13:44
• Maybe it sounds like ARIMA, but it ain't. ARIMA is easier to estimate because the forecast refers to previous real Y which is fixed. In my problem you refer to previous forecasted Y which changes anytime the parameters of the model change. – Przemyslaw Remin Mar 2 '12 at 10:20
• Your link does not work. Please provide the referred material in a form that enables users to see it if it is important for understanding your problem - i.e. perfectly data and the code in the body of your question. – Tim Mar 15 '15 at 20:15

you can easily estimate this model using R function which is optim() or using simple lm() function in iterative manner, first you suppress Y(t-1) by setting a=0 then estimate b and c, this is first iteration, the next step is to add Y(t-1) as independent variable and estimate Y(t) once again, and again put Y(t) as Y(t-1); notice that by Y(t) I meant fitted variable in vector notation