The Census method II 2 is an extension and refinement of the simple adjustment method. Subsequently, the term X has become synonymous with this refined version of the Census method II.

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In addition to the documentation that can be obtained from the Census Bureau, a detailed summary of this method is also provided in Makridakis, Wheelwright, and McGee and Makridakis and Wheelwright If you plot those data, it is apparent that 1 there appears to be an upwards linear trend in the passenger loads over the years, and 2 there is a recurring pattern or seasonality within each year i.

The purpose of seasonal decomposition and adjustment is to isolate those components, that is, to de-compose the series into the trend effect, seasonal effects, and remaining variability. The "classic" technique designed to accomplish this decomposition was developed in the 's and is also known as the Census I method see the Census I overview section. This technique is also described and discussed in detail in Makridakis, Wheelwright, and McGee , and Makridakis and Wheelwright The difference between a cyclical and a seasonal component is that the latter occurs at regular seasonal intervals, while cyclical factors usually have a longer duration that varies from cycle to cycle.

The trend and cyclical components are customarily combined into a trend-cycle component TC t. X t represents the observed value of the time series at time t. Consider the difference between an additive and multiplicative seasonal component in an example: The annual sales of toys will probably peak in the months of November and December, and perhaps during the summer with a much smaller peak when children are on their summer break. Thus, you could add to your forecasts for every December the amount of 3 million to account for this seasonal fluctuation.

The previous example can be extended to illustrate the additive and multiplicative trend-cycle components. In terms of the toy example, a "fashion" trend may produce a steady increase in sales e.

## How To Identify Patterns in Time Series Data: Time Series Analysis

In addition, cyclical components may impact sales. To reiterate, a cyclical component is different from a seasonal component in that it usually is of longer duration, and that it occurs at irregular intervals. The basic method for seasonal decomposition and adjustment outlined in the Basic Ideas and Terms topic can be refined in several ways. In fact, unlike many other time-series modeling techniques e.

Some of the major refinements are listed below. Trading-day adjustment.

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Different months have different numbers of days, and different numbers of trading-days i. When analyzing, for example, monthly revenue figures for an amusement park, the fluctuation in the different numbers of Saturdays and Sundays peak days in the different months will surely contribute significantly to the variability in monthly revenues. The X variant of the Census II method allows the user to test whether such trading-day variability exists in the series, and, if so, to adjust the series accordingly. Extreme values. Most real-world time series contain outliers, that is, extreme fluctuations due to rare events.

For example, a strike may affect production in a particular month of one year. Such extreme outliers may bias the estimates of the seasonal and trend components. The X procedure includes provisions to deal with extreme values through the use of "statistical control principles," that is, values that are above or below a certain range expressed in terms of multiples of sigma , the standard deviation can be modified or dropped before final estimates for the seasonality are computed.

Multiple refinements. The refinement for outliers, extreme values, and different numbers of trading-days can be applied more than once, in order to obtain successively improved estimates of the components.

The X method applies a series of successive refinements of the estimates to arrive at the final trend-cycle, seasonal, and irregular components, and the seasonally adjusted series. Tests and summary statistics. In addition to estimating the major components of the series, various summary statistics can be computed.

For example, analysis of variance tables can be prepared to test the significance of seasonal variability and trading-day variability see above in the series; the X procedure will also compute the percentage change from month to month in the random and trend-cycle components. As the duration or span in terms of months or quarters for quarterly X increases, the change in the trend-cycle component will likely also increase, while the change in the random component should remain about the same.

The width of the average span at which the changes in the random component are about equal to the changes in the trend-cycle component is called the month quarter for cyclical dominance , or MCD QCD for short. These and various other results are discussed in greater detail below. The computations performed by the X procedure are best discussed in the context of the results tables that are reported. The adjustment process is divided into seven major steps, which are customarily labeled with consecutive letters A through G.

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Prior adjustment monthly seasonal adjustment only. Before any seasonal adjustment is performed on the monthly time series, various prior user- defined adjustments can be incorporated. The user can specify a second series that contains prior adjustment factors; the values in that series will either be subtracted additive model from the original series, or the original series will be divided by these values multiplicative model.

For multiplicative models, user-specified trading-day adjustment weights can also be specified.

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These weights will be used to adjust the monthly observations depending on the number of respective trading-days represented by the observation. Preliminary estimation of trading-day variation monthly X and weights. Next, preliminary trading-day adjustment factors monthly X only and weights for reducing the effect of extreme observations are computed. Final estimation of trading-day variation and irregular weights monthly X- The adjustments and weights computed in B above are then used to derive improved trend-cycle and seasonal estimates.

These improved estimates are used to compute the final trading-day factors monthly X only and weights. Final estimation of seasonal factors, trend-cycle, irregular, and seasonally adjusted series. The final trading-day factors and weights computed in C above are used to compute the final estimates of the components. Modified original, seasonally adjusted, and irregular series.

The original and final seasonally adjusted series, and the irregular component are modified for extremes. The resulting modified series allow the user to examine the stability of the seasonal adjustment.

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In this part of the computations, various summary measures see below are computed to allow the user to examine the relative importance of the different components, the average fluctuation from month-to-month quarter-to-quarter , the average number of consecutive changes in the same direction average number of runs , etc.

Finally, you will compute various charts graphs to summarize the results. For example, the final seasonally adjusted series will be plotted, in chronological order, or by month see below. Customarily, these tables are numbered, and also identified by a letter to indicate the respective part of the analysis. For example, table B 11 shows the initial seasonally adjusted series; C 11 is the refined seasonally adjusted series, and D 11 is the final seasonally adjusted series.

Shown below is a list of all available tables. Also, for quarterly adjustment, some of the computations outlined below are slightly different; for example instead of a term [monthly] moving average, a 4-term [quarterly] moving average is applied to compute the seasonal factors; the initial trend-cycle estimate is computed via a centered 4-term moving average, the final trend-cycle estimate in each part is computed by a 5-term Henderson average. Following the convention of the Bureau of the Census version of the X method, three levels of printout detail are offered: Standard 17 to 27 tables , Long 27 to 39 tables , and Full 44 to 59 tables.

For the charts, two levels of detail are available: Standard and All. Distributed lags analysis is a specialized technique for examining the relationships between variables that involve some delay. For example, suppose that you are a manufacturer of computer software, and you want to determine the relationship between the number of inquiries that are received, and the number of orders that are placed by your customers.

You could record those numbers monthly for a one-year period, and then correlate the two variables. Put another way, there will be a time lagged correlation between the number of inquiries and the number of orders that are received. Time-lagged correlations are particularly common in econometrics.

For example, the benefits of investments in new machinery usually only become evident after some time. Higher income will change people's choice of rental apartments, however, this relationship will be lagged because it will take some time for people to terminate their current leases, find new apartments, and move. In general, the relationship between capital appropriations and capital expenditures will be lagged, because it will require some time before investment decisions are actually acted upon. In all of these cases, we have an independent or explanatory variable that affects the dependent variables with some lag.

## How To Identify Patterns in Time Series Data: Time Series Analysis

The distributed lags method allows you to investigate those lags. Detailed discussions of distributed lags correlation can be found in most econometrics textbooks, for example, in Judge, Griffith, Hill, Luetkepohl, and Lee , Maddala , and Fomby, Hill, and Johnson In the following paragraphs we will present a brief description of these methods. We will assume that you are familiar with the concept of correlation see Basic Statistics , and the basic ideas of multiple regression see Multiple Regression.

Suppose we have a dependent variable y and an independent or explanatory variable x which are both measured repeatedly over time. In some textbooks, the dependent variable is also referred to as the endogenous variable, and the independent or explanatory variable the exogenous variable.