We will now see how we can fit an MA model to a given time series using the `arima()`

function in R. Recall that MA model is an `ARIMA(0, 0, 1)`

model.

We can use the `arima()`

function in R to fit the MA model by specifying `order = c(0, 0, 1)`

.

We will perform the estimation using the `msft_ts`

time series that we created earlier in the first lesson. If you don’t have the `msft_ts`

loaded in your R session, please follow the steps to create it as specified in the first lesson.

Let’s start by creating a plot of the original data using the `plot.ts()`

function.

> plot.ts(msft_ts, main="MSFT prices", ylab="Prices")

We will fit the MA model to this data using the following command:

msft_ma <- arima(msft_ts, order = c(0, 0, 1))

The output contains many things including the estimated slope (ma1), mean (intercept), and innovation variance (sigma^2) as shown below:

> msft_ma Call: arima(x = msft_ts, order = c(0, 0, 1)) Coefficients: ma1 intercept 0.8602 55.2735 s.e. 0.0259 0.2603 sigma^2 estimated as 4.952: log likelihood = -559.83, aic = 1125.66 >

The `msft_ma`

object also contains the residuals (ε_{t} ). We can extract the residuals, using the `residuals()`

function in R.

residuals <- residuals(msft_ma)

Once you find the residuals ε_{t}, the fitted values are just X̂t=X_{t}−ε_{t}. In R, we can do it as follows:

msft_fitted <- msft_ts - residuals

We can now plot both the original and the fitted time series to see how close the fit is.

ts.plot(msft_ts) points(msft_fitted, type = "l", col = 2, lty = 2)

As we can see, the model does not provide a good fit for the original series. One possible reason is that our original series was not stationary. We can fit the ARIMA model with first order differencing by passing the parameter I=1.

> msft_ma <- arima(msft_ts, order = c(0, 1, 1)) > residuals <- residuals(msft_ma) > msft_fitted <- msft_ts - residuals > ts.plot(msft_ts) > points(msft_fitted, type = "l", col = 2, lty = 2)

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