Time-Series Analysis and Forecasting
Chapter Goals
After completing this chapter, you should be able to:
§ Compute and interpret index numbers
§ Weighted and unweighted price index
§ Weighted quantity index
§ Test for randomness in a time series
§ Identify the trend, seasonality, cyclical, and irregular components in a time series
§ Use smoothing-based forecasting models, including moving average and exponential smoothing
§ Apply autoregressive models and autoregressive integrated moving average models
Index Numbers
§ Index numbers allow relative comparisons over time
§ Index numbers are reported relative to a Base Period Index
§ Base period index = 100 by definition
§ Used for an individual item or measurement
Single Item Price Index
Consider observations over time on the price of a single item
§ To form a price index, one time period is chosen as a base, and the price for every period is expressed as a percentage of the base period price
§ Let p0 denote the price in the base period
§ Let p1 be the price in a second period
§ The price index for this second period is
Index Numbers: Example
§ Airplane ticket prices from 1995 to 2003:
Index Numbers: Interpretation
§ Prices in 1996 were 90% of base year prices
§ Prices in 2000 were 100% of base year prices (by definition, since 2000 is the base year)
§ Prices in 2003 were 120% of base year prices
Aggregate Price Indexes
§ An aggregate index is used to measure the rate of change from a base period for a group of items
Unweighted
Aggregate Price Index
Unweighted Aggregate Price Index: Example
§ Unweighted total expenses were 18.8% higher in 2004 than in 2001
Weighted
Aggregate Price Indexes
Laspeyres Price Index
Laspeyres Quantity Index
The Runs Test for Randomness
§ The runs test is used to determine whether a pattern in time series data is random
§ A run is a sequence of one or more occurrences above or below the median
§ Denote observations above the median with “+” signs and observations below the median with “-” signs
The Runs Test for Randomness
§ Consider n time series observations
§ Let R denote the number of runs in the sequence
§ The null hypothesis is that the series is random
§ Appendix Table 14 gives the smallest significance level for which the null hypothesis can be rejected (against the alternative of positive association between adjacent observations) as a function of R and n
The Runs Test for Randomness
§ If the alternative is a two-sided hypothesis on nonrandomness,
§ the significance level must be doubled if it is less than 0.5
§ if the significance level, a, read from the table is greater than 0.5, the appropriate significance level for the test against the two-sided alternative is 2(1 – a)
Counting Runs
Runs Test Example
§ Use Appendix Table 14
§ n = 18 and R = 6
§ the null hypothesis can be rejected (against the alternative of positive association between adjacent observations) at the 0.044 level of significance
§ Therefore we reject that this time series is random using a = 0.05
Runs Test: Large Samples
§ Given n > 20 observations
§ Let R be the number of sequences above or below the median
Consider the null hypothesis H0: The series is random
§ If the alternative hypothesis is positive association between adjacent observations, the decision rule is:
Runs Test: Large Samples
Consider the null hypothesis H0: The series is random
§ If the alternative is a two-sided hypothesis of nonrandomness, the decision rule is:
Example: Large Sample
Runs Test
§ A filling process over- or under-fills packages, compared to the median
Example: Large Sample
Runs Test
§ A filling process over- or under-fills packages, compared to the median
§ n = 100 , R = 45
Example: Large Sample
Runs Test
Time-Series Data
§ Numerical data ordered over time
§ The time intervals can be annually, quarterly, daily, hourly, etc.
§ The sequence of the observations is important
§ Example:
Year: 2001 2002 2003 2004 2005
Sales: 75.3 74.2 78.5 79.7 80.2
Time-Series Plot
§ the vertical axis measures the variable of interest
§ the horizontal axis corresponds to the time periods
Time-Series Components
Trend Component
§ Long-run increase or decrease over time (overall upward or downward movement)
§ Data taken over a long period of time
Trend Component
§ Trend can be upward or downward
§ Trend can be linear or non-linear
Seasonal Component
§ Short-term regular wave-like patterns
§ Observed within 1 year
§ Often monthly or quarterly
Cyclical Component
§ Long-term wave-like patterns
§ Regularly occur but may vary in length
§ Often measured peak to peak or trough to trough
Irregular Component
§ Unpredictable, random, “residual” fluctuations
§ Due to random variations of
§ Nature
§ Accidents or unusual events
§ “Noise” in the time series
Time-Series Component Analysis
§ Used primarily for forecasting
§ Observed value in time series is the sum or product of components
§ Additive Model
§ Multiplicative model (linear in log form)
Smoothing the Time Series
§ Calculate moving averages to get an overall impression of the pattern of movement over time
§ This smooths out the irregular component
Moving Average: averages of a designated
number of consecutive
time series values
(2m+1)-Point Moving Average
§ A series of arithmetic means over time
§ Result depends upon choice of m (the number of data values in each average)
§ Examples:
§ For a 5 year moving average, m = 2
§ For a 7 year moving average, m = 3
§ Etc.
§ Replace each xt with
Moving Averages
§ Example: Five-year moving average
§ First average:
§ Second average:
§ etc.
Example: Annual Data
…
Calculating Moving Averages
§ Each moving average is for a consecutive block of (2m+1) years
Annual vs. Moving Average
§ The 5-year moving average smoothes the data and shows the underlying trend
Centered Moving Averages
§ Let the time series have period s, where s is even number
§ i.e., s = 4 for quarterly data and s = 12 for monthly data
§ To obtain a centered s-point moving average series Xt*:
§ Form the s-point moving averages
§ Form the centered s-point moving averages
Centered Moving Averages
§ Used when an even number of values is used in the moving average
§ Average periods of 2.5 or 3.5 don’t match the original periods, so we average two consecutive moving averages to get centered moving averages
Calculating the
Ratio-to-Moving Average
§ Now estimate the seasonal impact
§ Divide the actual sales value by the centered moving average for that period
Calculating a Seasonal Index
Calculating Seasonal Indexes
Interpreting Seasonal Indexes
§ Suppose we get these seasonal indexes:
Exponential Smoothing
§ A weighted moving average
§ Weights decline exponentially
§ Most recent observation weighted most
§ Used for smoothing and short term forecasting (often one or two periods into the future)
Exponential Smoothing
§ The weight (smoothing coefficient) is a
§ Subjectively chosen
§ Range from 0 to 1
§ Smaller a gives more smoothing, larger a gives less smoothing
§ The weight is:
§ Close to 0 for smoothing out unwanted cyclical and irregular components
§ Close to 1 for forecasting
Exponential Smoothing Model
Exponential Smoothing Example
§ Suppose we use weight a = .2
Sales vs. Smoothed Sales
§ Fluctuations have been smoothed
§ NOTE: the smoothed value in this case is generally a little low, since the trend is upward sloping and the weighting factor is only .2
Forecasting Time Period (t + 1)
§ The smoothed value in the current period (t) is used as the forecast value for next period (t + 1)
§ At time n, we obtain the forecasts of future values, Xn+h of the series
Exponential Smoothing in Excel
§ Use tools / data analysis /
exponential smoothing
§ The “damping factor” is (1 – a)
§ To perform the Holt-Winters method of forecasting:
§ Obtain estimates of level and trend Tt as
§ Where a and b are smoothing constants whose values are fixed between 0 and 1
§ Standing at time n , we obtain the forecasts of future values, Xn+h of the series by
§ Assume a seasonal time series of period s
§ The Holt-Winters method of forecasting uses a set of recursive estimates from historical series
§ These estimates utilize a level factor, a, a trend factor, b, and a multiplicative seasonal factor, g
§ The recursive estimates are based on the following equations
§ After the initial procedures generate the level, trend, and seasonal factors from a historical series we can use the results to forecast future values h time periods ahead from the last observation Xn in the historical series
§ The forecast equation is
Autoregressive Models
§ Used for forecasting
§ Takes advantage of autocorrelation
§ 1st order – correlation between consecutive values
§ 2nd order – correlation between values 2 periods apart
§ pth order autoregressive model:
Autoregressive Models
§ Let Xt (t = 1, 2, . . ., n) be a time series
§ A model to represent that series is the autoregressive model of order p:
§ where
§ g, f1 f2, . . .,fp are fixed parameters
§ et are random variables that have
§ mean 0
§ constant variance
§ and are uncorrelated with one another
Autoregressive Models
§ The parameters of the autoregressive model are estimated through a least squares algorithm, as the values of g, f1 f2, . . .,fp for which the sum of squares
is a minimum
Forecasting from Estimated Autoregressive Models
§ Consider time series observations x1, x2, . . . , xt
§ Suppose that an autoregressive model of order p has been fitted to these data:
§ Standing at time n, we obtain forecasts of future values of the series from
§ Where for j > 0, is the forecast of Xt+j standing at time n and for j £ 0 , is simply the observed value of Xt+j
Autoregressive Model:
Example
Autoregressive Model:
Example Solution
Autoregressive Model Example: Forecasting
Autoregressive Modeling Steps
§ Choose p
§ Form a series of “lagged predictor” variables xt-1 , xt-2 , … ,xt-p
§ Run a regression model using all p variables
§ Test model for significance
§ Use model for forecasting
Chapter Summary
§ Discussed weighted and unweighted index numbers
§ Used the runs test to test for randomness in time series data
§ Addressed components of the time-series model
§ Addressed time series forecasting of seasonal data using a seasonal index
§ Performed smoothing of data series
§ Moving averages
§ Exponential smoothing
§ Addressed autoregressive models for forecasting