**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 p_{0} denote the price in the base period

§ Let p_{1} 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 H_{0}: 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 H_{0}: 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 x_{t} 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 X_{t}^{*:}

§ 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, X_{n+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 T_{t} 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, X_{n+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 X_{n} 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

§ p^{th} order autoregressive model:

Autoregressive Models

§ Let X_{t} (t = 1, 2, . . ., n) be a time series

§ A model to represent that series is the autoregressive model of order p:

§ where

§ g, f_{1} f_{2}, . . .,f_{p} are fixed parameters

§ e_{t} 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, f_{1} f_{2}, . . .,f_{p} for which the sum of squares

is a minimum

Forecasting from Estimated Autoregressive Models

§ Consider time series observations x_{1}, x_{2}, . . . , x_{t}

§ 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 X_{t+j} standing at time n and for j £ 0 , is simply the observed value of X_{t+j}

Autoregressive Model:

Example

Autoregressive Model:

Example Solution

Autoregressive Model Example: Forecasting

Autoregressive Modeling Steps

§ Choose p

_{§ }Form a series of “lagged predictor” variables x_{t-1} , x_{t-2} , … ,x_{t-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