6.4 The success criteria and process

The following elaborates on success criteria and the process

6.4.1 Success Criteria

Ultimately we want to be able to answer the following:

  1. Do we have trend?
  2. Do we have cyclical movements?
  3. Do we have seasons?
  4. Do we have autocorellation (elaborated in section ??)? If yes:
    1. If RHO = 1, then we can take first differences
    2. If RHO <> 1, then we can do the generalized differences., thus implies the following:
      1. Do an OLS and get the residuals
      2. Use the residuals in the following equation \(e_t=\rho e_{t-1}+\sigma_t\), using OLS as estimated rho (\(\hat{\rho}\))
        1. If rho = 0, then 0 autocorrelation

6.4.2 The Process

Deseasonalizing and detrending based on moving averages and accounting for cyclical moves

Note, if you do not have seasonality, then jump to section 2. trend etc.

6.4.2.1 Desaesonalizing

  1. Remove the short-term fluctuations
  2. If we have even number of periods, one must center the data. Whith odd period numbers, you can merely center with the period in the middle. The procedure with even number of periods is the following:

1. Find the MA_t with equation (6.1) and (6.2)

\[\begin{equation} MA_t = \frac{(Y_{t-2}+Y_{t-1}+Y_{t}+Y_{t+1}+)}{4} \tag{6.1} \end{equation}\] and

\[\begin{equation} MA_{t+1} = \frac{(Y_{t-1}+Y_{t}+Y_{t+1}+Y_{t+2}+)}{4} \tag{6.2} \end{equation}\]

Note, that the MA for each is centered in the center and rounded up to the coming period

2. Then find the centered MA

Then do the average of the two periods, which will find the actual center:

\[\begin{equation} CMA_t=\frac{(MA_t+MA_{t+1})}{2} \tag{6.3} \end{equation}\]

Notice, that the example is with quarterly MA, hence the MAs are divided with 4, this could have been monthly and then onw would divide by 12.

  1. \(CMA_t\) is representing the depersonalized data.
  2. One can find the seasonal factor by saying

\[\begin{equation} SF_t=\frac{Y_t}{CMA_t} \tag{6.4} \end{equation}\]

This explains whether one as above or below the expect season level.

  1. CONCLUSION: IF \(SF_t > 1\), then Y is greater than the quarterly (or what other period is used) average or, \(SF_t < 1\), then the Y is less than the quarterly average


Alternative, deseasonalizing data can be done by dividing the raw data with some seasonal index, that is adding dummy variables for the periods. Although by using the index, one assume, that the same seasonality is the same as preivous periods.


6.4.2.2 Long-term trend

Long-term trend, this is estimated from the deseasonalized data. This is estimated using simple linear regression. Basically the detrended data consists of the residuals between the actual data and the estimated data by using the trend variable (the counter 1 to n).

Task 1

We must find out if the trend is linear or quadratic.

Linear: \(C\hat{M}A_t=f(t)=\beta_0 + \beta_1t\)

Quadratic: \(C\hat{M}A_t=f(t)=\beta_0 + \beta_1t+\beta_2t^2\)

Where \(t\) is the time indicator and 1 = the first observation and increases by 1 thereafter.

Now we have obtained the centered moving-average trend

\[\begin{equation} CMAT = C\hat{M}A \tag{6.5} \end{equation}\]

6.4.2.3 Cyclical Component

Cyclical component, one can compare the CMA with the CMAT to find the cyclical factor. Thus, the cyclical factor is:

\[\begin{equation} CF = \frac{CMA}{CMAT} (\#CF) \end{equation}\]

If CF > 1, the deseasonalised value is above the long-term trend of the data. If the opposite, then below.

6.4.2.4 Time-Series decomposition forecast

Now we can do the reverse procedure, using the factors, that we have just found.

The reverse procedure is assembling the predicted Y based on the factors that have just been found.

\[\begin{equation} \hat{Y}=CMAT*SI*CF*I \tag{6.6} \end{equation}\]

Where,

  • CMAT = T,
  • S is the SF,
  • CF is the CF and
  • I is the irregular component (this is assumed to be 1 given its random nature, if one expects a boom or shock, this can be modeled with)

6.4.3 Autocorellation

ALWAYS ALWAYS CHECK FOR AUTOCORRELATION.

What to do?

  • You are missing some variable, find the missing variable(s). In practice this can be very difficult.
  • Do differencing
  • Use autoregressive model approach, where you are using lagged variables as variables to predict the coming period
    • We are going to talk about ARIMA (The box and jenkins methodology, more about this in section 7)

If our residuals have autocorelation, it means that there is some relationship in the model, that our model does not account four. In worst case, we can end up ‘proving’ some relationships between variables, that are in fact not true, but it is rather autocorellation that is proving the model, and the not the relationship between the variables and the dependent variables.