ARIMA equation interpretation

I'm trying to replicate ARIMA (1,0,1)(1,0,1) equation in excel as a formula but I am not able to understand the interpretation of white noise residual e(t) or u(t).If could help me understand the interpretation and how to derive it from raw data it'll be really great.

Thanks

• Just to clarify: are you trying to simulate or estimate the model? – Graeme Walsh Jan 8 '15 at 4:19
• I want it for estimation.For example,if I provide it with new data,say 5 more data points the Excel formula uses the new data to provide forecasts for the next 4-5 points – USer123 Jan 8 '15 at 5:01

1 Answer

Let's define an ARIMA(1,0,1)(1,0,1) model for a time series $y_t$ with period $s$ as:

$$(1-\phi_1L)(1-\phi_2L^s) y_t = (1+\theta_1L)(1+\theta_2L^s) \epsilon_t \,, \quad t=s+1,s+2,...,n \,,$$ where $L$ is the lag operator such that $L^iy_t = y_{t-i}$.

For your purposes, it is helpful to write the model above as follows:

$$y_t = \phi_1 y_{t-1} + \phi_2 y_{t-s} - \phi_1\phi_2 y_{t-s-1} + \epsilon_t + \theta_1 \epsilon_{t-1} + \theta_2 \epsilon_{t-s} + \theta_1\theta_2 \epsilon_{t-s-1} \,.$$

If you are comparing your results with some source, make sure that you are using the same definition of the model. Sometimes a different specification that changes the sign of the AR and MA coefficients is used.

The term $\epsilon_t$ is generally defined as a set of independent and identically distributed variables with mean equal to zero and variance $\sigma^2$, i.e., $\epsilon_t \sim IID(0, \sigma^2)$.

When forecasting, you deal with this term as follows:

• If the time period of the lagged $\epsilon_t$ belongs to the observed sample period, then the estimate of $\epsilon_t$ is evaluated as the difference between the value observed for $y_t$ at that period and the estimated value of $y_t$ for the same period.

• If the time period of the white noise is outside the sample period, then the estimate of the white noise is replaced by its expectation, defined above as zero.

In the first case you need to get the values estimated for the series at each period of the sample (the fitted values). For this, you may find useful this post and this post. Basically, you can apply a recursive procedure starting from the first observation. At each iteration, the equation of the ARIMA model defined above is evaluated for the observed values and given the parameters of the model. The terms $\epsilon_t$ are handled as above, the lags are replaced by the error terms obtained from previous iterations and the current value $\epsilon_t$ by its expectation, zero. For simplicity you may start the recursion by setting the first values of $\epsilon_t$ to zero.

• Thanks I tried the same on my own on (0,0,1) model but it fails on (1,0,1)(1,0,1) model – USer123 Jan 13 '15 at 13:43
• @USer123 I have edited my answer and changed the equation of the model. My previous definition of the model may have confused you since it was an ARMA(S,0,0) with zero coefficients for the lags from 2 to S-1. Now the model is the seasonal ARMA you are interested in. See the indexing and all the terms involved in the model that become more clear when it is rewritten after multiplying the regular and the seasonal polynomials. – javlacalle Jan 13 '15 at 19:19
• Thanks, this equation has Constant/intercept missing. I'm facing issue calculating intercept/constant. the constant given by R or SPSS does not work in the equation – USer123 Jan 22 '15 at 4:30
• Depending on how the model is defined--with intercept/constant or mean--you may need to use the model specification $y_t = \alpha + \phi_1 y_{t-1} + \cdots + \epsilon_t + \theta\epsilon_{t-1} + \cdots$ or $(y_t - \alpha) = \phi_1 (y_{t-1} - \alpha) + \cdots + \epsilon_t + \theta\epsilon_{t-1} + \cdots$. – javlacalle Jan 22 '15 at 9:22