2Types of DataTime series data: a sequence of observations measured over time (usually at equally spaced intervals, e.g., weekly, monthly and annually). Examples of time series data include:Gross Domestic Product each quarter;annual rainfall;daily stock market indexCross sectional data: data on one or more variables collected at the same point in time
3Time Series vs Causal Modeling Causal (regression) models: the investigator specifies some behavioural relationship and estimates the parameters using regression techniques;Time series models: the investigator uses the past data of the target variable to forecast the present and future values of the variable
4Time Series vs Causal Modeling On the other hand, there are many cases when one cannot, or one prefers not to, build causal models:insufficient information is known about the behavioural relationship;lack of, or conflicting, theories;insufficient data on explanatory variables;expertise may be unavailable;time series models may be more accurate
5Time Series vs Causal Modeling Direct benefits of using time series models:Little storage capacity is needed;some time series models are automatic in that user intervention is not required to update the forecasts each period;some time series models are evolutionary in that the models adapt as new information is received;
6Classical Decomposition of Time Series Trend – does not necessarily imply a monotonically increasing or decreasing series but simply a lack of constant mean, though in practice, we often use a linear or quadratic function to predict the trend;Cycle – refers to patterns or waves in the data that are repeated after approximately equal intervals with approximately equal intensity. For example, some economists believe that “business cycles” repeat themselves every 4 or 5 years;
7Classical Decomposition of Time Series Seasonal – refers to a cycle of one year duration;Random (irregular) – refers to the (unpredictable) variation not covered by the above
8Decomposition Method Multiplicative Models Additive Models Find the estimates of these four components.
9Multiplicative Decomposition Examples:(1) US Retail and Food Services Sales from Q1 to 2008 Q1Figure 2.1(2) Quarterly Number of Visitor Arrivals in Hong Kong from 2002 Q1 to 2008 Q1Figure 2.2
17Assuming the average of the observations is also the median of the observations, the MA for periods 1 – 4, 2 – 5, 3 – 6 are centered at positions 2.5, 3.5 and 4.5 respectively.
18To get an average centered at periods 3, 4, 5 etc To get an average centered at periods 3, 4, 5 etc. the means of two consecutive moving averages are calculated:Centered MovingAverage for period 3Average for period 4
22After all have been computed, they are further averaged to eliminate irregularities in the series. We also adjust the seasonal indices so that they sum to the number of seasons in a year (i.e., 4 for quarterly data, 12 for monthly data). Why?)
41Multiplicative decomposition is used when the time series exhibits increasing or decreasing seasonal variation (Yt=TCt SNt IRt)TCtSNtYtYt – Yt-1Yr 1Q1Q2Q3Q411.51314.522.214.171.124.217.256.511.619.2–10.755.17.6Yr 217.51920.52226.259.516.426.4–16.756.910
42Additive decomposition is used when the time series exhibits constant seasonal variation (Yt=TCt + SNt + IRt)TCtSNtYtYt – Yt-1Yr 1Q1Q2Q3Q411.51314.5161.8–1–1.50.713.31216.7–1.313.7Yr 217.51920.52219.31822.7
43Step 1 : Estimation of seasonal component (SNt) Calculation of MAt and CMAt is the same as per multiplicative decompositionInitial seasonal component may be estimated byFor example,
44Seasonal indices are averaged and adjusted so that they sum to zero (Why?)