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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