Time Series Analysis

1. Classical Decomposition – additive or multiplicative effects  Trend – long run  Seasonal – periodic business patterns (months, weeks, days)  Cyclical – business conditions (recessions, inflations)  Error – randomness

Use in Bureau of Labor Statistics: http://www.bls.gov/lau/laumthd.htm Wikipedia: http://en.wikipedia.org/wiki/Decomposing_of_time_series

Data set 1977 births in New York state http://wweb.uta.edu/faculty/eakin/busa3321/1977NewYorkStatebirths.xls

2. Models

2.1 Descriptive

 Exponential Smoothing – giving recent values different weights than old values

Used in Forecasting costs: http://www.dtic.mil/dtic/tr/fulltext/u2/a483271.pdf search for exponential smoothing

 Moving Averages (average values before and after a date)

Used in the minutes of the Federal Open Market Committee: http://www.federalreserve.gov/monetarypolicy/fomcminutes20090318.htm

 Index numbers – ratio of current value to a value in a specific previous time period

Used in costs trends: http://www.usbr.gov/tsc/techreferences/mands/cct.html

2.2 Inferential

2.2.1 Least Squares Models

2.2.1.1 Trend Models

2.2.1.1.1 Simple linear models

 Measuring the effect of one variable on another over time or  The effect of time on the variable 2.2.1.1.1 Exponential Trend Models

Log10(Yt) = 0 + 1Xt or Loge(Yt) = 0 + 1Xt

Coefficient interpretation: For each one-unit increase in X, the estimated average value of Y increases by 101

Example if predicted log sales = 10 + 0.0015(advertising), then

(10)0.0015= 1.0035

For each dollar of advertising, the estimated average sales increase by 0.35%.

Mentioned in Business Week: http://www.businessweek.com/the_thread/economicsunbound/archives/20 09/01/after_the_crisi.html

2.2.1.2 Seasonal Effects

 Use dummy variables to represent the season.

 For example, for quarterly data use 3 dummy variables

2.2.1.3 Cyclical Effects

Determine leading indicators of business conditions

2.2.1.4 Problems with Least Squares in Time Series Data

2.2.1.4.1 Positively correlated errors due to sampling over time.

2.2.1.4.1.1 Estimated standard errors too small – All inferences are invalid.

2.2.1.4.1.2 Detection: Durbin-Watson test

If calculated DW value is small than a table value, significant positive correlated errors.

2.2.1.4.1.3 Remedy:

o Estimate and remove positive correlation

o First, Second, and Percentage differences Discussed in an addendum of the Federal Trade Association http://www.fta.dot.gov/printer_friendly/planning_environment_2427 .html

2.2.1.4.2 Unequal variance

Remedy – Transform dependent variable

2.2.1.4.3 Predicting Outside Range of data

No way to travel into the future.

2.2.2 Autoregressive Moving Average Models

 Use to estimate correlation of errors over time

 Beyond scope of notes

 Mentioned in a Bureau of Labor Statistics report: http://www.bls.gov/cpi/cpisahoma.htm