Time series analysis part 2

Winters' Exponential Smoothing Method: When is it used?

Winters' exponential smoothing method is a forecasting method intended to take into account both trend and seasonal variation. There are two versions of Winters' method: additive and multiplicative.

Step 1: Find the seasonal indexes for additive 1. Moving averages 2. Centered moving averages (if even number of data points) 3. subtract CMAs from correct values 4. medians 5. indexes (additional normalization) 6. interpretation

1. First compute a four-period moving average MA(4). (We are using 4 data values for the moving average because we have quarterly data; with monthly data, 12 data values would have been used.) In terms of the time sequence, the first entry can be thought of as occurring between the second and third data values. The second entry can be thought of as occurring between the third and forth data values, etc. 2. We want numbers that are centered on actual data values so we do a second moving average called a centered moving average (CMA): The first entry for CMA averages the first two four-period moving averages. In terms of the time sequence, this average is centered on the third time period By proceeding in this manner, we end up with a moving average that smoothes out the seasonal variations in the data and has values centered on an actual data point in the time sequence. Because the CMA smoothes out the seasonal variation, what is left is the trend component, the cyclical component and the random error, Tt + Ct + It 3. When we subtract the CMA from the correct y value (THIS MEANS STARTING BY SUBTRACTING FROM THE THIRD Y VALUE FOR QUARTLY DATA OR FROM THE 7TH Y VALUE FOR MONTHLY DATA, AND THEN SUBTRACT FROM EVERY CONSEQUTIVE Y VALUE AFTERWARDS), this essentially removes the trend from the sales data, so each entry in this column consists only of the effect of the seasonal component and the random error. 4. Then we collect these (Sales - CMA) values for like quarters and find the medians. 5. An additional normalization is made to the medians so that they SUM TO ZERO. This is achieved by subtracting the average of the 4 medians from each median. 6. The normalized medians are the seasonal indexes for each quarter and show the adjustment to any forecast that needs to be made to allow for seasonal effects. For example, if the 1st quarter index is 30, this means that quarter 1 sales are 30 dollars above the annual average.

3 options for fixing autocorrelation if present missing variable transformation lagged value

1. If autocorrelation is due to the omission of an important variable from the regression, the remedy to this problem is to locate the missing variable and include it in the model, although this is easier said than done in many practical situations. 2. Use appropriate transformations of the original time series variables. The most commonly used estimation procedures are the Prais-Winsten method, full maximum likelihood method and Cochrane-Orcutt method. 3. Add a lagged value of the response variable as an explanatory variable. This is a viable option if the number of observations is reasonably large since one observation is lost because of the lagged variable

Positive vs negative autocorrelation

1. Positive autocorrelation With positive first-order autocorrelation, a positive residual is followed by a positive residual in the next period, and a negative residual is followed by another negative residual. The positive autocorrelation is the most frequent type of autocorrelation in business applications. 2. Negative autocorrelation With negative first-order autocorrelation, we expect a positive residual to be followed by a negative residual, followed by a positive residual, and so on. Negative autocorrelation is very rarely present in business and economic data.

Three approaches for dealing with seasonality:

1. Regression with dummy variables for the seasons 2. Deseasonalizing data 3. Winters' exponential smoothing method All three approaches have their strengths and weaknesses and there is no point in arguing which of these methods is best. When they are applied properly, they have all been found to be useful in real business situations.

The Durbin-Watson test is not reliable for sample sizes smaller than

15

Durbin-Watson statistic and interpretation

A numerical measure has been developed to check for lag 1 autocorrelation. It is called the Durbin-Watson statistic The Durbin-Watson statistic can have a value ranging from 0 to 4. A value of 2 indicates no autocorrelation. Typically values below 2 indicate positive autocorrelation, and values above 2 indicate negative autocorrelation.

When to use additive vs multiplicative time series

Additive models are typically used when the seasonal component is constant, that is the amplitudes of the seasonal cycles are roughly the same. Multiplicative seasonal models are typically used when the seasonal variation is growing over time, that is the amplitudes of the seasonal cycles are proportional to the level of the time series. In practice, it is recommended to fit both and use the accuracy measures to help determine which method produces the most accurate forecasts.

Step 3: Fit an appropriate forecasting model to the deseasonalized data (same for additive and multiplicative)

Because the data have been deseasonalized, we can think of these numbers as consisting of a trend component and random error. Therefore, we can apply a linear, quadratic, or exponential trend regression

How to detect autocorrelation

One way to detect autocorrelation is to plot the residuals in time order. If a positive autocorrelation effect exists, there will be clusters of residuals with the same sign, and you will readily detect an apparent pattern. If negative autocorrelation exists, residuals will tend to jump back and forth from positive to negative to positive, and so on.

Can other variables be used in the regression equation in addition to the seasonal indicator variables?

Other variables could be used in the regression in addition to the seasonal indicator variables. The other variables included in the regression equation could be time t, non-linearly transformed t and x variables, lagged versions of t y , current or lagged versions of other explanatory variables. These variables would capture any time series behavior other than seasonality.

Deseasonalizing Data 5 Steps

Step 1: Find the seasonal indexes Step 2: Deseasonalize the original data Step 3: Fit an appropriate forecasting model to the deseasonalized data Step 4: Calculate the desired forecasts Step 5: Reseasonalize the forecasts In Step 3, any of the methods we have studied (e.g. trend methods, single and double exponential smoothing techniques, moving averages method) can be used other than regression with dummies for the seasons and Winters' method, which wouldn't make much sense here because the data has been deseasonalized.

Step 1: Find the seasonal indexes for multiplicative 3. divide correct values by CMAs (ratio to moving averages) 4. means 5. indexes (additional normalization) 6. interpretation

Steps 1 and 2 are the same. 3. Then we DIVIDE the correct y value for each quarter by the corresponding centered moving average. THIS MEANS STARTING BY DIVIDING THE THIRD Y VALUE FOR QUARTLY DATA OR THE 7TH Y VALUE FOR MONTHLY DATA, AND THEN DIVIDE EVERY CONSEQUTIVE Y VALUE AFTERWARDS. These values are called the ratios-to-moving-average. 4. Next we compute the mean ratio-to-moving-average value for each season (that is, each quarter's (if using quarters) or month's (if using months) ratio-to-moving-average is averaged over the years.) 5. The sum of the means should equal the number of the seasons (4 in the case of quarterly data, 12 in the case of monthly). Therefore, an additional normalization is made to the means so that they sum to the number of seasons. This is achieved by dividing each mean by the average of the 4 (in the case of quarterly) means. 6. The normalized means are the seasonal indexes for each qurter and can be interpreted as a percentage above or below the average for the year. For example, the seasonal index for the winter quarter is 1.4420. This indicates that sales during the winter are 44.20% above the average for the year.

what are seasonally unadjusted forecasts?

The forecasts calculated from the regression on the deseasonalized data.

dummy variable interpretation What do they mean? What should you do if any are insignificant?

The interpretation of the coefficients of the dummy variables is the same as for any set of dummy variables. They represent the differences between the fitted values for the indicated quarter and the reference (base-level) quarter. If the coefficients of any of the dummies turn out to be statistically insignificant, they can be omitted from the equation. Then the omitted terms are effectively combined with the reference season. For example, if the 1 Q term were omitted, then quarters 1 and 4 would essentially be combined and treated as the reference season, and the other two seasons would be compared to them through their dummy variable coefficients.

Seasonal component of time series (and how to detect)

The seasonal component of a time series represents those changes (highs and lows) that occur at approximately the same time every year. The easiest way to determine whether there is seasonality in a time series is to check its graph and see if it has a regular pattern of ups and/or downs in particular months or quarters. Although random noise can sometimes mask such a pattern, the seasonal pattern is usually fairly obvious.

Auto correlation

The second assumption in regression analysis states that the error terms are probabilistically independent. This assumption is often violated for time series data. The problem with time series data is that the residuals are often correlated with nearby residuals because an event in one time period may influence an event in the next time period. We call this undesirable property autocorrelation (or serial correlation).

Step 2: Deseasonalize the original data for multiplicative model

To deseasonalize an observation, we divide it by the respective seasonal index

Step 4: Calculate the desired forecasts (same for additive and multiplicative)

To find a forecast value, we use the trend line regression equation. These are SEASONALLY UNADJUSTED forecasts.

Step 5. Reseasonalize the forecasts for additive model

To find the forecasts for the original time series, we adjust the seasonally unadjusted forecasts by ADDING the seasonal indexes

Step 2: Deseasonalize the original data for additive model

To obtain the deseasonalized values, we subtract the seasonal indexes from all the original data (matching the index season to each data points season)

Step 5. Reseasonalize the forecasts for multiplicative model

We do this by MULTIPLYING the unadjusted forecast values by the appropriate seasonal index.

In the note example, if the Durbin Watson stat was between Dl and Du, what was concluded?

no definite decision can be made BUT ALSO although the time series graph shows some signs of positive autocorrelation, it is not enough to cause a serious concern.

This method of detecting seasonality by using dummy variables in a multiple regression model is always an option. Just remember that the number of dummy variables is always...

one less than the number of seasons. If the data are quarterly, then three dummies are needed; if the data are monthly, then 11 dummies are needed.

The seasonally unadjusted forecasts with their respective seasonal index added or multiplied back in are called?

seasonally adjusted forecasts.

If the number of data values used for a moving average is odd, the moving average will be associated with the time period of _______. In such cases, we would not have to _________ because _____________________

the middle observation. center the moving average; the moving averages would already be associated with one of the time periods from the original time series.