In your report, comment on at least 4 of the following: [
Report on the validity of the analysis.]
 The relevance and usefulness of your forecast (who might use the information and for what purpose)
 The reliability of the forecasts (how good they are) with justification.
 Two further features of the time-series data in addition to the trend.
 Appropriateness of, and possible improvements to, the model you used in your analysis.
 Possible improvements to your model.
 The limitations of your analysis.
 Seasonally adjusted data together with an interpretation of these data.
 Potential sources of bias
The above bullet points are the Excellence component of the Achievement Standard. You are expected to go into detail about a variety of these issues ensuring there are no major contradictions or statistical misunderstandings. You cannot duplicate points and all comments must be justified in context. Comments of a trivial or highly speculative nature are not acceptable.
NOTE: At least one of your Excellence comments must relate to your data or your analysis.
The following notes explain the above points in a little more detail.
Relevance and usefulness of the forecast; Reliability of the forecast
The closer the period of the forecast to the data from which it is made, the more reliable and useful the forecast will be. Conversely, the further ahead in time the forecast is, the less reliability it has since the conditions in which the original data were obtained may not continue to hold.
The usefulness of the forecast may also be limited by the variability within the data. This may be evident if the smoothed data (moving averages) do not lie close to the fitted trend line (so if a least squares trend line has been found, the R 2 value will be relatively small) and hence there is considerable variation in the individual effects for each season.
Another way to examine how accurate your forecast may be is to create a forecasted series for the whole time period, allowing a visual comparison between the raw data and your predictions – if you forecasted series closely follows the raw data then your future forecast is likely to be more reliable.
Relevance and reliability may also be affected by the amount of data available. If the time series spans a short time overall (eg quarterly data collected for three consecutive years), there will not be many individual seasonal effects for each season. When the mean of the individual seasonal effects for a particular season (eg the March quarter) is found, the value will be based on limited information and will not be particularly robust.

The usefulness of a forecast may not be evident if the purpose of carrying out the analysis is not known. Even if the purpose is known, there may be very important and compelling reasons why the "owner" or user of the data needs the forecast but which are not evident or as compelling to others – for example, if the results are to contribute to decisions regarding major capital expenditure - and this makes it difficult to make informed or constructive comment
Examples of appropriate evidence presented by students for assessment for this achievement standard may include comments relating to:
 the proximity of the forecast period in relation to that of the data evidence gathered
 reliability and variability in the data
 data quantity but this must be more significant than comments such as “limited data available” and must relate to the number of periods that are covered by the data
 the purpose of the data and the use that will be made of the forecast.
Features of the time series
Features of the time series that may be noteworthy include the following. Comments must be in context, and wherever possible (and sensible) include numbers to explain your point.
Some of these points may also fit into the Improvements to your model category.
 How much variability is there? Does the variability appear constant or does it change over the period from which the data were collected? If there is an apparent change in the variability, is it increasing or decreasing? It should be noted that it is likely that any observations about variability within the data are being made on a limited number of observations.

Are there any outliers? Can any plausible reason for the outliers be identified (eg a particularly wet season may depress farm production for a particular quarter)?
 Can the trend for the series be modelled by one line (straight or curvilinear) over the duration of the series or is it more appropriate to use two or more models over different ranges within the time series?
 What is the seasonal pattern evident in the series? Is it constant across the series, or does it change over time?

Does there appear to be an identifiable change in the underlying nature of the data at some point? If so, can it be accounted for – ie can a potential cause be identified (eg a change in a trend might be caused by a sudden change in the popularity of and subsequent demand for a particular commodity)?
Potential sources of bias
If comment is being made on data that has been collected by someone else, it can be difficult to be realistic about potential sources of bias, and at best they are often simply conjecture. It is also necessary to consider the purpose of the analysis and in what way(s) the findings will be used. For example, if data are collected from one region of the country and a statistical inference is made about the whole country, there may be no bias if previous similar studies have revealed that the region from which the data were collected is representative of the whole country and there is no reason to believe that there has been no change that could cause bias. Or if the stakes are not high (eg the results are not going

to be used to make decisions about major expenditure), then the experimental design may allow for some potential bias because using a more sophisticated design to reduce it may not be cost-effective.
In most well-conducted experiments (and this would apply to most data sets from reputable sources), the experimental design should minimise potential sources of bias.
For students to make valid comments on this aspect of their investigation, they would need to have been supplied with detailed information about the purpose(s) of collecting the data, the data source and the data collection method(s).
Improvements to the model
Improvements could include using a multiplicative model for the series instead of an additive model or modelling the trend using more than one regression line over different ranges of the data (if suggested by the nature of the data).
An additive model for time series data X t
is of the form X t
= T + S + E, where T represents the trend component, S the seasonal component and E the random or error effects (note that the model ignores any cyclic effect). A multiplicative model for time series data X t
is of the form X t
= T x S x E.
An additive model is appropriate if the graph of the time series shows
 a generally uniform overall increase or decrease in the data (ie what is probably a linear trend), and/or
 a generally uniform variability of the data about the observed trend.
A multiplicative model is appropriate if the graph of the time series shows
 a non-linear overall increase or decrease in the data (ie what is probably a non-linear trend), and/or
 increasing or decreasing variability of the data about the observed trend.
Direct comparisons may be drawn with the graphs of arithmetic (additive) and geometric
(multiplicative) sequences as shown below. Note that the time series are representations only, and show quarterly data with the quarters numbered 1, 2, 3 and 4 for reference.
Arithmetic sequence t n
= a + (n-1)d Geometric sequence t n
= ar n-1

Time series X t
= T + S + E Time series X t
= T x S x E
Deciding whether to use an additive or a multiplicative model requires judgement (which develops through experience).
Limitations of the analysis
Limitations could include the following.

Range of data. If data are available only over a limited range, then only a small number of values are available to estimate each seasonal effect, for example. Any comments made by students need to be more significant than a simple remark such as "having more data would have improved the reliability of the forecast". This may have been
covered under a different point.

The method of analysis. When using moving averages, equal weighting is given to data at the start of the period as is given to data at the end of the period. However, it may be more appropriate when making forecasts to give greater (and differing) weights to the most recent data as these values could be expected to be better determinants of future performance, particularly in the immediate future. In addition, using moving averages results in some of the data values at the end (and the beginning) of the time series not having corresponding moving averages so that these values are unable to produce individual seasonal effects. These are general limitations with analyses involving moving averages, and other methods of analysing time series have been developed to circumvent them. Comments made by students need to be related to the particular investigation rather than being general remarks.
Seasonally adjusted data
From the 2006 National Moderators Report