Time Series Forecasting: Objectives

• Gain a general understanding of time series forecasting techniques.

• Understand the four possible components of time-series data.• Understand stationary forecasting techniques.• Understand how to use regression models for trend analysis.• Learn how to decompose time-series data into their various

elements and to forecast by using decomposition techniques.• Understand the nature of autocorrelation and how to

test for it.• Understand auto-regression in forecasting.

Time-Series Forecasting

• Time-series data: data gathered on a given characteristic over a period of time at regular intervals

• Time-series techniqueso Attempt to account for changes over time by

examining patterns, cycles, trends, or using information about previous time periods

o Naive Methodso Averagingo Smoothingo Decomposition of time series data

Time Series Components

• Trend – long term general direction, typically 8 to 10 years

• Cycles (Cyclical effects) – patterns of highs and lows through which data move over time periods usually of more than a year, typically 3 to 5 years

• Seasonal effects – shorter cycles, which usually occur in time periods of less than one year.

• Irregular fluctuations – rapid changes or “bleeps” in the data, which occur in even shorter time frames than seasonal effects.

Time-Series Effects

Time Series Components

• Stationary time-series - data that contain no trend, cyclical, or seasonal effects.

• Error of individual forecast et – the difference between the actual value xt and the forecast of that value Ft i.e.

Measurement of Forecasting Error

• Error of the Individual Forecast (et = Xt – Ft) is the difference between the actual value xt and the forecast of that value Ft.

• Mean Absolute Deviation (MAD) - is the mean, or average, of the absolute values of the errors.

• Mean Square Error (MSE) - circumvents the problemof the canceling effects of positive and negative forecast errors. – Computed by squaring each error and averaging the

squared errors.

Measurement of Forecasting Error

• Mean Percentage Error (MPE) – average of the percentage errors of a forecast

• Mean Absolute Percentage Error (MAPE) – average of the absolute values of the percentage errors of a forecast

• Mean Error (ME) – average of all the errors of forecast for a group of data

Nonfarm Partnership Tax Returns: Actual and Forecast with = .7

Year Actual Forecast Error1 14022 1458 1402.0 56.03 1553 1441.2 111.84 1613 1519.5 93.55 1676 1584.9 91.16 1755 1648.7 106.37 1807 1723.1 83.98 1824 1781.8 42.29 1826 1811.3 14.7

10 1780 1821.6 -41.611 1759 1792.5 -33.5

Mean Absolute Deviation (MAD): Nonfarm Partnership Forecasted Data

Year Actual Forecast Error |Error|1 1402.02 1458.0 1402.0 56.0 56.03 1553.0 1441.2 111.8 111.84 1613.0 1519.5 93.5 93.55 1676.0 1584.9 91.1 91.16 1755.0 1648.7 106.3 106.37 1807.0 1723.1 83.9 83.98 1824.0 1781.8 42.2 42.29 1826.0 1811.3 14.7 14.7

10 1780.0 1821.6 -41.6 41.611 1759.0 1792.5 -33.5 33.5

674.5

Mean Square Error (MSE): Nonfarm Partnership Forecasted Data

Year Actual Forecast Error Error2

1 14022 1458 1402.0 56.03 1553 1441.2 111.84 1613 1519.5 93.55 1676 1584.9 91.16 1755 1648.7 106.37 1807 1723.1 83.98 1824 1781.8 42.29 1826 1811.3 14.7

10 1780 1821.6 -41.611 1759 1792.5 -33.5

55864.2

3136.012499.2

8749.78292.3

11303.67038.51778.2

214.61731.01121.0

Smoothing Techniques

• Smoothing techniques produce forecasts based on “smoothing out” the irregular fluctuation effects in the time-series data

• Naive Forecasting Models - simple models in which it is assumed that the more recent time periods of data represent the best predictions or forecasts for future outcomes

Smoothing Techniques

• Averaging Models - the forecast for time periodt is the average of the values for a given number of previous time periods:o Simple Averageso Moving Averageso Weighted Moving Averages

• Exponential Smoothing - is used to weight data from previous time periods with exponentially decreasing importance in the forecast.

Simple Average Model

The forecast for time period t is the average of the values for a given number of previous time periods.

Month Year

Cents per

Gallon Month Year

Cents per

GallonJanuary 2 61.3 January 3 58.2February 63.3 February 58.3March 62.1 March 57.7April 59.8 April 56.7May 58.4 May 56.8June 57.6 June 55.5July 55.7 July 53.8August 55.1 August 52.8September 55.7 SeptemberOctober 56.7 OctoberNovember 57.2 NovemberDecember 58.0 December

The monthly average last12 months was 56.45,

so I predict56.45 for September.

Moving Average

• Updated (recomputed) for every new time period• May be difficult to choose optimal number of periods• May not adjust for trend, cyclical, or seasonal effects

Update each period.

Demonstration Problem 15.1:Four-Month Moving Average

Shown in the following table here are shipments(in millions of dollars) for electric lighting and wiring equipment over a 12-month period. Use these datato compute a 4-month moving average for all available months.

Demonstration Problem 15.1:Four-Month Moving Average

Months Shipments

4-Mo Moving Average

Forecast Error

January 1056February 1345March 1381April 1191May 1259 1243.25 15.75June 1361 1294.00 67.00July 1110 1298.00 -188.00August 1334 1230.25 103.75September 1416 1266.00 150.00October 1282 1305.25 -23.25November 1341 1285.50 55.50December 1382 1343.25 38.75

Demonstration Problem 15.1:Four-Month Moving Average

Weighted Moving AverageForecasting Model

A moving average in which some time periods are weighted differently than others.

Example of 3 monthsWeighted average

where last month’s value

value for the previous month

value for the month before the previous monthThe denominator = the total of weights

1tM

2tM

3tM

Demonstration Problem 15.2:Four-Month Weighted Moving Average

Months Shipments

4-Month WeightedMoving Average

Forecast Error

January 1056February 1345March 1381April 1191May 1259 1240.88 18.13June 1361 1268.00 93.00July 1110 1316.75 -206.75August 1334 1201.50 132.50September 1416 1272.00 144.00October 1282 1350.38 -68.38November 1341 1300.50 40.50December 1382 1334.75 47.25

Exponential Smoothing

Used to weight data from previous time periods with exponentially decreasing importance in the forecast

t t t

t

t

t

F X FFFX

where

1

1

1

: the forecast for the next time period (t+1)

the forecast for the present time period (t)

the actual value for the present time period= a value between 0 and 1

is the exponentialsmoothing constant

Demonstration Problem 15.3: = 0.2

The U.S. Census Bureau reports the total units of new privately owned housing started over a 16-year recent period in the United States are given here. Use exponential smoothing to forecast the values for each ensuing time period. Work the problem using = 0.2, 0.5, and 0.8

Demonstration Problem 15.3: = 0.2 = 0.2

YearHousing Units

(1,000) F e |e| e2

1990 1193 -- -- -- --1991 1014 1193.0 -179 179 320411992 1200 1157.2 42.8 42.8 18321993 1288 1165.8 122.2 122.2 149331994 1457 1190.2 266.8 266.8 71182

1995 1354 1243.6 110.4 110.4 121881996 1477 1265.7 211.3 211.3 446481997 1474 1307.9 166.1 166.1 275891998 1617 1341.1 275.9 275.9 761211999 1641 1396.3 244.7 244.7 598782000 1569 1445.2 123.8 123.8 153262001 1603 1470.0 133.0 133.0 176892002 1705 1496.6 208.4 208.4 434312003 1848 1538.3 309.7 309.7 959142004 1956 1600.2 355.8 355.8 1265942005 2068 1671.4 396.6 396.6 157292

3146.5 796657

MAD 209.8

MSE 53110

Demonstration Problem 15.3: = 0.8

= 0.8

YearHousing Units

(1,000) F e |e| e2

1990 1193 -- -- -- --1991 1014 1193.0 -179 179 64.01992 1200 1049.8 150.2 150.2 3770.01993 1288 1170.0 118.0 118.0 29832.21994 1457 1264.4 192.6 192.6 27736.91995 1354 1418.5 -64.5 64.5 21114.61996 1477 1366.9 110.1 110.1 44970.21997 1474 1455.0 19.0 19.0 49023.41998 1617 1470.2 146.8 146.8 20083.91999 1641 1587.6 53.4 53.4 13535.82000 1569 1630.3 -61.3 61.3 36967.32001 1603 1581.3 21.7 21.7 4166.22002 1705 1598.7 106.3 106.3 12120.02003 1848 1683.7 164.3 164.3 361.72004 1956 1815.1 140.9 140.9 21551.32005 2068 1927.8 140.2 140.2 6140.4

1668.3 228896

MAD 111.2

MSE 15245.9

Trend Analysis

• Trend – long run general direction of climate overan extended time

• Linear Trend• Quadratic Trend• Holt’s Two Parameter Exponential Smoothing -

Holt’s technique uses weights (β) to smooth the trend in a manner similar to the smoothing used in single exponential smoothing (α)

Average Hours Worked per Weekby Canadian Manufacturing Workers

Following table provides the data needed tocompute a quadratic regression trend model onthe manufacturing workweek data

Average Hours Worked per Weekby Canadian Manufacturing Workers

Period Hours Period Hours Period Hours Period Hours1 37.2 11 36.9 21 35.6 31 35.72 37.0 12 36.7 22 35.2 32 35.53 37.4 13 36.7 23 34.8 33 35.64 37.5 14 36.5 24 35.3 34 36.35 37.7 15 36.3 25 35.6 35 36.56 37.7 16 35.9 26 35.67 37.4 17 35.8 27 35.68 37.2 18 35.9 28 35.99 37.3 19 36.0 29 36.0

10 37.2 20 35.7 30 35.7

Excel Regression Output using Linear Trend

Regression StatisticsMultiple RR SquareAdjusted R SquareStandard ErrorObservations

ANOVASS MS F Significance F

Regression 1 13.4467 13.4467 51.91 .00000003Residual 33 8.5487 0.2591Total 34 21.9954

Coefficients Standard Error t Stat P-valueIntercept 37.4161 0.17582 212.81 .0000000Period -0.0614 0.00852 -7.20 .00000003

i ti i

t

Y X

X

where

Y

0 1

37 416 0 0614

:

. .

data value for period i

time periodi

i

YX

35

0.7820.6110.56000.509

df

Excel Regression Output usingQuadratic Trend

Regression StatisticsMultiple R 0.8723R Square 0.761Adjusted R Square 0.747Standard Error 0.405Observations 35

ANOVAdf SS MS F Significance F

Regression 2 16.7483 8.3741 51.07 1.10021E-10Residual 32 5.2472 0.1640Total 34 21.9954

Coefficients Standard Error t Stat P-valueIntercept 38.16442 0.21766 175.34 2.61E-49Period -0.18272 0.02788 -6.55 2.21E-07Period2 0.00337 0.00075 4.49 8.76E-05

i ti ti i

ti

t t

Y X X

XX X

where

Y

0 1 2

2

2

238164 0183 0 003

:

. . .

data value for period i

time period

the square of the i period

i

i

th

YX

Graph of Canadian Workweek Data with a Second-Degree Polynomial Fit

Demonstration Problem 15.4

Data on the employed U.S. civilian labour force (in 100,000) for 1991 through 2008, obtained from the U.S. Bureau of Labor Statistics. Use regression analysis to fit a trend line through the data and explore a quadratic trend. Compare the models.

Regression Output from Package

Model Comparison

Linear Model

Quadratic Model

Time Series: Decomposition

Decomposition – Breaking down the effects of time series data into four component parts: trend, cyclical, seasonal, and irregular

Basis for analysis is the Multiplicative Model Y = T · C · S · I

where:T = trend componentC = cyclical componentS = seasonal componentI = irregular component

Household Appliance Shipment Data

Illustration of decomposition process: the 5-year quarterly time-series data on U.S. shipments of household appliances

Year Quarter Shipments Year Quarter Shipments 1 1 4009 4 1 4595

2 4321 2 47993 4224 3 44174 3944 4 4258

2 1 4123 5 1 42452 4522 2 49003 4657 3 45854 4030 4 4533

3 1 44932 48063 45514 4485

Shipments in $1,000,000.

Development of Four-QuarterMoving Averages

Quarter Shipments4 Qtr M.T. 2 Yr M.T.

4 Qtr Centered

M.A.

Ratios of Actual Values to M.A.

1 1 40092 4321 16,4983 4224 16,612 33,110 4139 102.06%4 3944 16,813 33,425 4178 94.40%

2 1 4123 17,246 34,059 4257 96.84%2 4522 17,332 34,578 4322 104.62%3 4657 17,702 35,034 4379 106.34%4 4030 17,986 35,688 4461 90.34%

3 1 4493 17,880 35,866 4483 100.22%2 4806 18,335 36,215 4527 106.17%3 4551 18,437 36,772 4597 99.01%4 4485 18,430 36,867 4608 97.32%

4 1 4595 18,296 36,726 4591 100.09%2 4799 18,069 36,365 4546 105.57%3 4417 17,719 35,788 4474 98.74%4 4258 17,820 35,539 4442 95.85%

5 1 4245 17,988 35,808 4476 94.84%2 4900 18,263 36,251 4531 108.13%3 45854 4533

S·I(100)

T·C

Ratios of Actual to Moving Averages

1 2 3 4 5Q1 96.84% 100.22% 100.09% 94.84%Q2 104.62% 106.17% 105.57% 108.13%Q3 102.06% 106.34% 99.01% 98.74%Q4 94.40% 90.34% 97.32% 95.85%

Eliminate the Max and Min for each Quarter

1 2 3 4 5Q1 96.84% 100.22% 100.09% 94.84%Q2 104.62% 106.17% 105.57% 108.13%Q3 102.06% 106.34% 99.01% 98.74%Q4 94.40% 90.34% 97.32% 95.85%

Eliminate the maximum and the minimum for each quarter to remove irregular fluctuations. Average the remaining ratios for each quarter.

Computation of Average of Seasonal Indexes

1 2 3 4 5 AverageQ1 96.84% 100.09% 98.47%Q2 106.17% 105.57% 105.87%Q3 102.06% 99.01% 100.53%Q4 94.40% 95.85% 95.13%

Deseasonalized House Appliance Data

Year QuarterShipments(T*C*S*I)

SeasonalIndexes

(S)

DeseasonalizedData

(T*C*I)1 1 4009 98.47% 4,071

2 4321 105.87% 4,081 3 4224 100.53% 4,202 4 3944 95.12% 4,146

2 1 4123 98.47% 4,187 2 4522 105.87% 4,271 3 4657 100.53% 4,632 4 4030 95.12% 4,237

3 1 4493 98.47% 4,563 2 4806 105.87% 4,540 3 4551 100.53% 4,527 4 4485 95.12% 4,715

4 1 4595 98.47% 4,666 2 4799 105.87% 4,533 3 4417 100.53% 4,393 4 4258 95.12% 4,476

5 1 4245 98.47% 4,311 2 4900 105.87% 4,628 3 4585 100.53% 4,561 4 4533 95.12% 4,765

Autocorrelation (Serial Correlation)

• Autocorrelation occurs in data when the error terms of a regression forecasting model are correlated and not independent, particularly with economic variables.

• Potential Problems• Estimates of the regression coefficients no longer have

the minimum variance property and may be inefficient.• The variance of the error terms may be greatly

underestimated by the mean square error value.• The true standard deviation of the estimated regression

coefficient may be seriously underestimated.• The confidence intervals and tests using the t and F

distributions are no longer strictly applicable.

Autocorrelation (Serial Correlation)

• First-order autocorrelation occurs when there is correlation between the error terms of adjacent time periods.

• If first-order autocorrelation is present, the error for one time period is a function of the error of the previous time period

• is the first order auto correlation coefficient that measures the correlation between the error terms.

Autocorrelation (Serial Correlation)

• lies between -1 and 0 and +1 like , coefficient of correlation

• is a normally distributed error term• If positive autocorrelation is present, the value of is

between 0 and +1• If negative autocorrelation is present, the value of is

between 0 and -1• If then 0 which means there is no autocorrelation

and is just a random independent error term

Durbin-Watson Test

HHa

0 00

::

D

t t

where

e e

et

n

tt

n

2

2

2

1

1

: n = the number of observations

If D > do not reject H (there is no significant autocorrelation).

If D < , reject H (there is significant autocorrelation).

If , the test is inconclusive.

U 0

L 0

L U

dd

d d

,

D

Autoregression Model• Second Order Autoregression Model with two lagged

variables

• Third Order Autoregression Model with three lagged variables

Overcoming Autocorrelation Problem

• Addition of Independent Variables• Transforming Variables

First-differences approach - Often autocorrelation occurs in regression analyses when one or more predictor variables have been left out of the analysis

Percentage change from period to period - each value of x is subtracted from each succeeding time period value of x; these “differences” become the new and transformed x variable, the same for y

Use autoregression - multiple regression technique in which the independent variables are time-lagged versions of the dependent variable