Time Series

Techniques and an Example from Roger Simon’s The City Building

Process

What is a Time Series?

• Numerical data ordered by intervals of time in chronological order

• The analysis of a time series involves decomposing the series into its components, e.g., – Trends– Cyclical and Seasonal

Fluctuations– Irregular Fluctuations

Example of a Time Series

1888 2

1889 29

1890 33

1891 40

1892 56

1893 43

1894 39

1895 21

1896 38

Etc… 31

To Analyze a Time Series…

• Identify the possible processes affecting the series:– Inflation or Deflation

(for series of prices or expenditures)

– Long term trend– Cyclical fluctuations– Seasonal fluctuations– Irregular fluctuations

Steps to Analyze Changing Construction Costs

• Identify a measure:– Construction Cost? Or…. – Construction Cost Per Square Foot

• Convert to Constant Dollars

• Measure the Trend

• Identify cyclical or seasonal factors

• Identify irregular fluctuations

Raw Construction Cost over Time

Properties Built over Time

Size of Buildings Built over Time

Adjusting for Inflation• if yrbuilt >887 and yrbuilt < 912 then let concost2=concost*20• if yrbuilt >911 and yrbuilt < 916 then let concost2=concost/.06• if yrbuilt = 916 then let concost2=concost/.065• if yrbuilt = 917 then let concost2=concost/.077• if yrbuilt = 918 then let concost2=concost/.090• if yrbuilt = 919 then let concost2=concost/.104• if yrbuilt = 920 then let concost2=concost/.120• if yrbuilt = 921 then let concost2=concost/.107• if yrbuilt = 922 then let concost2=concost/.103• if yrbuilt = 923 then let concost2=concost/.102• if yrbuilt = 924 then let concost2=concost/.102• if yrbuilt = 925 then let concost2=concost/.105• if yrbuilt = 926 then let concost2=concost/.106• if yrbuilt = 927 then let concost2=concost/.104• if yrbuilt = 928 then let concost2=concost/.102• if yrbuilt = 929 then let concost2=concost/.102• if yrbuilt=. then let concost2= concost *20• if concost=. Then let concost2 =.

Construction Cost Adjusted for Inflation

Comparing Raw and Inflation Adjusted Construction Costs

Construction Cost per Square Foot (Inflation Adjusted)

Construction Cost per Square Foot (Inflation Adjusted)

The Model: Cost per square foot = constant(a) + b1*year +

b2*(last year’s cost per square foot)

Dep Var: COSTPER N: 39 Multiple R: 0.885 Squared multiple R: 0.784 Adjusted squared multiple R: 0.772 Standard error of estimate: 4.143 Effect Coefficient Std Error Std Coef Tolerance t P(2 Tail)

CONSTANT 7.330 3.100 0.000 . 2.364 0.024LAGCOSTP 0.589 0.148 0.540 0.325 3.974 0.000YEAR2 0.293 0.103 0.387 0.325 2.851 0.007

Analysis of VarianceSource Sum-of-Squares df Mean-Square F-ratio P Regression 2244.126 2 1122.063 65.375 0.000Residual 617.887 36 17.164

Dep Var: COSTPER N: 39 Multiple R: 0.885 Squared multiple R: 0.784 Adjusted squared multiple R: 0.772 Standard error of estimate: 4.143 Effect Coefficient Std Error Std Coef Tolerance t P(2 Tail)

CONSTANT 7.330 3.100 0.000 . 2.364 0.024LAGCOSTP 0.589 0.148 0.540 0.325 3.974 0.000YEAR2 0.293 0.103 0.387 0.325 2.851 0.007

Analysis of VarianceSource Sum-of-Squares df Mean-Square F-ratio P Regression 2244.126 2 1122.063 65.375 0.000Residual 617.887 36 17.164

The Model: Cost per square foot = constant(a) + b1*year +

b2*last year’s cost per square foot

• Cost per square foot = $7.33 + .59 (last year’s cost) + .29 (year)

• Number of cases: 39 (1889-1928). 1889=1

• Adjusted R Square: 77.2

Trend and Residuals

YearCost Per

Square FootLast Year's Cost

Per Sq. Ft.PredictedCost PerSq. Ft.

Residual

889 $21.91 $22.67 $20.99 0.92890 $20.42 $21.91 $20.83 -0.41891 $17.27 $20.42 $20.25 -2.98892 $17.40 $17.27 $18.68 -1.29893 $19.21 $17.40 $19.05 0.16894 $19.76 $19.21 $20.41 -0.65895 $20.85 $19.76 $21.03 -0.19896 $19.36 $20.84 $21.96 -2.60897 $20.73 $19.36 $21.38 -0.65898 $20.54 $20.73 $22.48 -1.94899 $25.83 $20.54 $22.66 3.17900 $28.09 $25.83 $26.08 2.01901 $28.80 $28.09 $27.70 1.10902 $28.63 $28.80 $28.41 0.22903 $33.00 $28.63 $28.61 4.40904 $31.60 $33.00 $31.48 0.12905 $34.47 $31.60 $30.94 3.53906 $32.93 $34.47 $32.93 0.00907 $35.72 $32.93 $32.32 3.40908 $37.37 $35.72 $34.25 3.12909 $37.97 $37.37 $35.52 2.45910 $34.66 $37.97 $36.17 -1.51911 $35.13 $34.66 $34.51 0.63912 $30.27 $35.13 $35.08 -4.81913 $37.89 $30.27 $32.51 5.38914 $28.78 $37.89 $37.29 -8.51915 $38.23 $28.78 $32.22 6.01916 $37.99 $38.23 $38.08 -0.09917 $40.33 $37.99 $38.23 2.10918 $23.85 $40.33 $39.91 -16.06919 $30.15 $23.85 $30.49 -0.34920 $33.45 $30.15 $34.49 -1.04921 $34.19 $33.45 $36.73 -2.54922 $34.37 $34.19 $37.46 -3.09923 $36.37 $34.37 $37.86 -1.49924 $36.81 $36.37 $39.33 -2.53925 $42.96 $36.81 $39.88 3.08926 $49.20 $42.96 $43.80 5.40928 $53.59 $49.20 $48.07 5.52

. $53.59 . .

The Data:1.Year 2. Cost Per Square Foot3. Last Year’s Cost Per Square Foot4. Predicted Cost5. Residual