1. ## LilDragonfly's Regression Q3

This is posted on behalf of LilDragonfly.

Attached is a data set called "Retail Footware Sales in New Zealand March 1991 to December 2000".

Use this data to answer the following questions:

1. Analyse this time series. This should include:
a) seasonally adjusting the data and interpreting the results in context
b) choosing a model to represent the recent trend using formal methods,
for example appropriate numerical calculations or least squares regressiosn
c) evaluating your model using residual analysis

2. Write a report on the analysis of this time series data. You should include:
the methods and results of your analysis and a discussion of the features of the series, how you could improve your model, limitations of your analsis.

2. Originally Posted by CaptainBlack
This is posted on behalf of LilDragonfly.

Attached is a data set called "Retail Footware Sales in New Zealand March 1991 to December 2000".

Use this data to answer the following questions:

1. Analyse this time series. This should include:
a) seasonally adjusting the data and interpreting the results in context

I will remove the seasonality by using a 9-point moving average, that is
the smoothed sales for period $\displaystyle i$ will be taken to be:

$\displaystyle SmoothdSales(i)=\left(\sum_{k=i-4}^{i-1} Sales(k)+\sum_{k=i+1}^{i+4}Sales(k) \right)/8$.

This gives the smoothed sales in the $\displaystyle i$-th quarter as the average
quarterly sales for the preceeding four quarters and the following four quarters.

That is the smoothing is taking place over an integral number of years and
so the seasonality is averaged out. Also note that this means that the
sales for quarter $\displaystyle i$ are not used in calculating the smoothed
sales for quarter $\displaystyle i$.

The results of this smoothing is shown in the section of an excel spreadsheet
shown in the attachment, and the graph attached.

The data shows seasonal behaviour with two peaks per year the first the
jun quarter and the second in the dec quarter. There is also a general
downdward trend in sales, but with substantial variation about that trend.

3. Originally Posted by CaptainBlack
This is posted on behalf of LilDragonfly.

Attached is a data set called "Retail Footware Sales in New Zealand March 1991 to December 2000".

Use this data to answer the following questions:

1. Analyse this time series. This should include:
[indent]a) seasonally adjusting the data and interpreting the results in context
b) choosing a model to represent the recent trend using formal methods,
for example appropriate numerical calculations or least squares regressions
c) evaluating your model using residual analysis

Selecting a model to represent the trend this data is problematical.
There is a downward trend but also a significant variation about
the trend line and the question is do we want to model that variation?

I would stick with a simple linear trend line (as shown in the plot attached
to an earlier post), this is calculated using the standard linear regression
method. I think that trying to model the variation about the trend seems
to me to be over modelling (though an additional sinusoidal component looks
as though it could represent a substantial part of the remaining systematic
variation in the data about the trend line).

There are a number of residual plots that could be employed in this analysis.
The most significant is a simple plot of the residual against quarter. This
is shown in the attachment.

The residual plot shows that the residuals are not randomly scattered about
the zero residual axis as would be the case if we had modelled all of the
systematic variation in the data.

RonL

4. Originally Posted by CaptainBlack
This is posted on behalf of LilDragonfly.

Attached is a data set called "Retail Footware Sales in New Zealand March 1991 to December 2000".

Use this data to answer the following questions:

2. Write a report on the analysis of this time series data. You should include:
the methods and results of your analysis and a discussion of the features of the series, how you could improve your model, limitations of your analsis.

Most of what could be said to answer this part of the question has been said
above, it probably only needs more elaboration here.

Some comment on long term economic cycles and effects of other external
factors could be relevant.

Look at your notes and/or examples to see exactly what is expected to
answer this part of the question. This sort of thing usually depends on why
the customer has commissioned the study.

RonL

5. The seasonality can be extracted by averaging the smoothed/actual
data for quarters that correspond to the same season. This gives seasonality
factors:

March.0.955742591
June.. 1.064359786
Sept...0.881293072
Dec....1.084365967

RonL

6. A more sophisticated model of the long term variation of this data can
be modelled as:

$\displaystyle sales(t)=a.t+b+c.\sin(2. \pi.f.t+\phi)$

This can be fitted to the de-seasoned data using Nonlinear Least Squares.
This has been done and is illurstrated in the attachment.

The parameter values for this model are:

Code:
a	-0.334988202
b	67.43608622
f	0.050651566
phi	2.469557225
c	2.464159211
The residual plot for this model can be seen in the secon attachment. As
can be seen this residual plot is much better.