time series dr. sevtap kestel albert ludwigs universität ... time... · 3.1 scatterplot and...
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Time Series Analysis
Dr. Sevtap Kestel
Albert‐Ludwigs‐Universität FreiburgDepartment of Economics
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1. Introduction
Lecture 1
2
Time Series AnalysisDr. Sevtap Kestel
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1.1.1 Descriptive Statistics
• Population and Sample– Finite and infinite population
• Data– Quantitative– Qualitative (nominal, ordinal)
• Variable– Discrete – Continuous
• Graphical representations– Bar graphs, pie charts,
histograms, ogive • Measures of location
– Mean– Median– Mode
• Measures of dispersion– Range– Absolute mean deviation– Variance and standard
deviation– The five‐number summary and
Boxplots
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2.1 Estimation Techniques2.1.1 Types of Estimation
Point EstimationPopulation mean
Population variance
MLE‐Maximum Likelihood EstimatorMax{∏n
i=1f(xi)}
Interval EstimationPopulation Mean
Population Variance
/ 2 / 2( ) 1P X z X zn nα ασ σμ α− ≤ ≤ + = −
1
1 n
ii
X xn =
= ∑
2 2
1
1 ( )1
n
ii
s x Xn =
= −− ∑
2 22
2 21 / 2 / 2
( 1) ( 1)( ) 1n s n sPα α
σ αχ χ−
− −≤ ≤ = −
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2.1.2 Properties of Estimation
• Unbiased
• Consistent
• Efficient
ˆ[ ]E θ θ=
ˆlim{ [ ]} 0n
Var θ→∞
=
1 2 1ˆ ˆ ˆ[ ] [ ]Var Var is efficientθ θ θ≤ →
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2.2 Hypothesis Testing
2.2.1. Types of Hypothesis• Simple TestsHo: θ=θ0 vs HA: θ=θ1
• Composite TestsHo: θ=θ0 vs HA: θ>θ1 one‐tailed testHo: θ=θ0 vs HA: θ<θ1 one‐tailed testHo: θ=θ0 vs HA: θ≠θ1 two tailed‐test
• Test Statistic is the measure
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Types of errors
• Type I error: Reject Ho when it is true
• Type II error: Reject HA when it is true
• Power of the test:
Prob. Rejecting H0 when θ=θ1• P‐value: Prob. of rejecting H0 under null hypothesis
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2.2.2 Testing Population Parameters
• Mean: Test Statistics– Large sample
– Small sample
• Variance
• Proportion
• Decision: Reject H0 when test statistics falls within the Rejection Region
0
/Xz
nμ
σ−
=
0
/Xts n
μ−=
22
20
( 1)n sχσ−
=
0
0 0
ˆ/
p pzp q n−
=
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3. „Linear Models“
9
Time Series AnalysisDr. Sevtap Kestel
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3.1 Scatterplot and Correlation
Relation between two random variables
Linear, quadratic, exponential etc.
Covariance, Cov(X,Y), is the average amount of interaction between two
variables
Correlation Coefficient, ρ,is the measure dependency among two
variables
xE[(X- )( )] ( , )ρ=( ) ( )
1 ρ 1ρ 1 perfect correlationρ=0 no correlation
Y
x Y
Y Cov X YVar X Var Y
μ μσ σ
−=
− ≤ ≤= ±
Scatterplot
40
50
60
70
80
90
100
150 160 170 180 190 200
height
wei
ght
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3.1 Sample Correlation Coeffcient
x2 2
x Y
2 2 2 2 2 2
E[(X- )( )] [ ] [ ] [ ]ρ=( ) ( ) [( ) ] [( ) ]
E[(X-X)( )]( ( ) ) ( ( ) )
( )( )
1
1 ρ 1ρ 1 perfect correlationρ=0 no correlation
Y
X Y
i i
x y
Y E XY E X E YVar X Var Y E X E Y
n XY X YY Yrs s X X Y Y
X X Y Ys s
rn
μ μμ μ
− −=
− −
−−= =
− −
− −
=−
− ≤ ≤=±
∑ ∑ ∑∑ ∑ ∑ ∑
∑
Height Weight
161 55
164 58
166 63
168 66
169 70
172 74
173 76
174 79
176 80
176 82
177 84
178 83
180 84
182 85
183 87
185 89
188 90
189 91
191 93
, 0.965weight heightr =
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3.2 Simple Linear Regression
• Aim: to fit a linear model to two or more variables
X: independent variable (explanatory)
Y: dependent variable (response)
• Simple Linear Regression
Y=a+bX+ random error
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3.2 Regression Analysis
Random error ε ~N(0,σ2)
Parameters: a,b, or ai, i=0,1,..,n, are estimated by
Least Square Estimation Technique (LSE)
LSE: min{Σerror2}
ˆResidual=ε=Y-Y=Actual-Estimate
Y
X
sˆ ˆa=Y-bX; b=rs
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Example:Suppose Mr. Bump observes the selling price and sales volume of
milk gallons for 10 randomly selected weeks as follows
Week
Weekly sales level, Y*
Selling price, X, $
1 10 1.30
2 6 2.0
3 5 1.70
4 12 1.50
5 10 1.60
6 15 1.20
7 5 1.60
8 12 1.40
9 17 1.00
10 20 1.10
* Thousand of gallons
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WEEK x y x^2 y^2 xy
1 1.3 10 1.69 100 13
2 2 6 4 36 12
3 1.7 5 2.89 25 8.5
4 1.5 12 2.25 144 18
5 1.6 10 2.56 100 16
6 1.2 15 1.44 225 18
7 1.6 5 2.56 25 8
8 1.4 12 1.96 144 16.8
9 1 17 1 289 17
10 1.1 20 1.21 400 22
sum 14.4 112 21.56 1488 149.3
Normal equation:
10
10
ˆ4.14ˆ)10(1120
ˆˆ
ββ
ββ
+=
+= ∑∑ ii xny
8.119)112)(4.14(149)10(
6.2331121488)10()(
24.8)4.14(56.21)10()(222
222
−=−=−=
=−=−=
=−=−=
∑∑∑∑∑∑∑
yxxynS
yynS
xxnS
xy
yy
xx
14.3210
4.14)54.14(10
112ˆˆ
54.1424.8
8.119)4.14()56.21)(10(
112)4.14()3.149)(10(ˆ
0
21
=−−=−=
−=−
=−
−==
xy
SS
xx
xy
ββ
β
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Regression model
xy 54.1414.32ˆ −=
86.01121488)10(4.1456.21)10(
112)4.14(3.149)10(22
−=−−
−==
yyxx
xy
SSS
r
Standard error of estimate:
week x
Actual y
Estimated y error error^2
1 1.3 10 13.238 -3.238 10.48464
2 2 6 3.06 2.94 8.6436
3 1.7 5 7.422 -2.422 5.866084
4 1.5 12 10.33 1.67 2.7889
5 1.6 10 8.876 1.124 1.263376
6 1.2 15 14.692 0.308 0.094864
7 1.6 5 8.876 -3.876 15.02338
8 1.4 12 11.784 0.216 0.046656
9 1 17 17.6 -0.6 0.36
10 1.1 20 16.146 3.854 14.85332
sum 14.4 112 112.024 0 59.4248272.2
842.59
2)ˆ(
2ˆ
22
==−
−=
−= ∑∑
nyy
nerror
eσ
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Predicting Y: Suppose Mr. Bump wished to forecast the quantity of
milk sold if the price were set at $1.63
44.863.1)54.14(14.32ˆ 63.1 =−== =xYEy
Standard error of the forecast measures when x=1.63 is
90.28240.0
)44.163.1(101172.2
)()(11
2
2
2
=−
++=−
−++=∑ xx
xxn i
ep σσ
Then 95% prediction interval is 8.44 ± tn‐2, 0.025 2.90 => 8.44±2.306(2.90) => (1.753, 15.121)
Standard error of estimator of β1 00.3824.072.2
)( 21==
−==∑ xxS
e
xx
e σσσβ
Hypothesis TestingH0: β=0 vs H0: β≠0 8.4
00.3054.14
−=−−
=t < ‐2.306 Reject Ho
Coefficient of determination746.0
6.23314.591
)()ˆ(
11 2
22 =−=
−
−−=−=∑∑
yyyy
SSSEr
i
ii
yy
Approximately 75% of Y is explained by X.
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4. „Linear Models“
18
Time Series AnalysisDr. Sevtap Kestel
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4. Outline
4.1 Multiple Linear Regression
4.2 Nonlinear Regression
4.3 Regression Techniques
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4.1 Multiple Linear Regression
• Multiple Linear Regression
Y=a0+a1X1 +a2X2+..+anXn+ random error
• Use of matrix algebra
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Example: Suppose Mr. Bump observes the selling price, sales volume of milk gallons and advertising expense for 10 randomly selected weeks as follows (* Thousand of gallons; **
hundreds of dollars)
Week sales level, Y*
Selling price, X1 , $
Advertising**, X2
1 10 1.30 9
2 6 2.0 7
3 5 1.70 5
4 12 1.50 14
5 10 1.60 15
6 15 1.20 12
7 5 1.60 6
8 12 1.40 10
9 17 1.00 15
10 20 1.10 21
Correlation Matrix
165.089.065.0186.0
89.086.01
−−−
−=
adspricesales
adspricesales
r
The model :
εβββ +++= 22110 XXY
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Scatter Diagram Y versus X1
0
5
10
15
20
25
0.5 1 1.5 2 2.5
price
sal
es
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weeky X1 X2 y^2 X1^2 X2^2 X1 y
X2y X1X2
1 10 1.3 9 100 1.69 81 13 90 11.7
2 6 2 7 36 4 49 12 42 14.0
3 5 1.7 5 25 2.89 25 8.5 25 8.5
4 12 1.5 14 144 2.25 196 18 168 21.0
5 10 1.6 15 100 2.56 225 16 150 24.0
6 15 1.2 12 225 1.44 144 18 180 14.4
7 5 1.6 6 25 2.56 36 8 30 9.6
8 12 1.4 10 144 1.96 100 16.8 120 14.0
9 17 1 15 289 1 225 17 255 15.0
10 20 1.1 21 400 1.21 441 22 420 23.1
sum
11214.
4 1488 21.56 1522.0 149.3
1480.0 155.3
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Normal equations:
∑∑∑∑∑∑∑∑
∑∑∑
++=
++=
++=
222211202
212211101
22110
ˆˆˆ
ˆˆˆ
ˆˆˆ
iiiii
iiiii
iii
xxxxyx
xxxxyx
xxny
βββ
βββ
βββ
1522ˆ3.155ˆ144ˆ1480
3.155ˆ56.21ˆ4.14ˆ3.149
114ˆ4.14ˆˆ10112
210
210
210
βββ
βββ
βββ
++=
++=
++=
In matrix form:
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=′
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
=
2
1
0
ˆ
ˆ
ˆ
ˆ
15223.1551143.15556.214.14
1144.1410
2110.11......700.21930.11
20..610
β
β
β
βXXXY
( )⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=′′=
′′=′
−
5851.02476.8
4064.16)(ˆ
)(ˆ
1 YXXX
XXYX
β
β
31 59.025.841.16ˆ xxy +−=
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51.1310
9.15)ˆ(ˆ
22
=−
=−
−=
−= ∑∑
knyy
knerror
eσ
Variables used to explain variance of Y
none 0 233.6
Price 0.75 59.4
Price and advertisement 0.93 15.9
week
Price x1
Advertising
x2
Actual y
Estimated y error error^2
1 1.3 9 10 10.95 -0.95 0.9032
2 2 7 6 4.01 1.99 3.972
3 1.7 5 5 5.31 -0.31 0.0967
4 1.5 14 12 12.22 -0.22 0.0512
5 1.6 15 10 11.98 -1.98 3.947
6 1.2 12 15 13.53 1.46 2.159
7 1.6 6 5 6.72 -1.72 2.961
8 1.4 10 12 10.71 1.28 1.622
9 1 15 17 16.93 0.064 0.0042
10 1.1 21 20 19.62 0.379 0.14355
Sum0 15.90
( )( ) 91248.0
9/6.2337/9.151
1/)(/)ˆ(
1ˆˆ
1 2
2
2
22 =−=
−−
−−−=−=∑∑
nyyknyy
ry
ec σ
σ
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4.2 Non‐Linear Regression
• Exponential Equation of best fit
• Power equation of best fit
xY =ablogY =loga+xlogb
bY = aXlo g Y = lo g a+ b lo g X
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A start‐up company has developed an improved electronic chip for use in laboratory equipment. The company needs to project the manufacturing cost, so it develops spreadsheet models that takes into
account the purchase of production equipment, overhead, raw materials, depreciation, maintenance and other business costs.
The spreadsheet estimates the cost of producing 10,000 to 200,000 chips per year.
Chips Produced (1000s)
Cost
per
chi
p
200150100500
150
125
100
75
50
Scatterplot of Cost per chip vs Chips Produced (1000s)
correlation = ‐0.823
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Prod Costfitted Cost residual
10 146.1 96.7941 49.3059
20 105.8 92.482 13.318
30 85.75 88.1699 -2.4199
40 77.02 83.8578 -6.8378
50 66.1 79.5457 -13.4457
60 63.92 75.2336 -11.3136
70 58.8 70.9215 -12.1215
80 50.91 66.6094 -15.6994
90 47.22 62.2973 -15.0773
100 44.31 57.9852 -13.6752
120 42.88 49.361 -6.481
140 39.05 40.7368 -1.6868
160 37.47 32.1126 5.3574
180 35.09 23.4884 11.6016
200 34.04 14.8642 19.1758
Fitted Value
Res
idua
l
100908070605040302010
50
40
30
20
10
0
-10
-20
Residuals Versus the Fitted Values(response is Cost per chip)
Chips Produced (1000s)
Res
idua
l
200150100500
50
40
30
20
10
0
-10
-20
R esiduals Versus Chips Produced (1 0 0 0 s)(response is Cost per chip)
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logprod
cost
5.55.04.54.03.53.02.52.0
150
125
100
75
50
Scatterplot of cost vs logprod
prod.
logc
ost
200150100500
5.00
4.75
4.50
4.25
4.00
3.75
3.50
Scatterplot of logcost vs prod.
logprod
logc
ost
5.55.04.54.03.53.02.52.0
5.00
4.75
4.50
4.25
4.00
3.75
3.50
Scatterplot of logcost vs logprod
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Correlations: logcost, logprod, prod., cost
Logcost Logprod prod
logprod -0.997
prod -0.920 0.923
logcost 0.975 -0.976 -0.823
Regression Analysis: logcost versus logprod The regression equation is
logcost = 6.16 ‐ 0.502 logprod
Predictor Coef SE Coef T PConstant 6.15922 0.04454 138.30 0.000logprod ‐0.50162 0.01034 ‐48.52 0.000
S = 0.0329768 R‐Sq = 99.5% R‐Sq(adj) = 99.4%
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logprod
logc
ost
5.55.04.54.03.53.02.52.0
5.00
4.75
4.50
4.25
4.00
3.75
3.50
Scatterplot of logcost vs logprod
Fitted Value
Stan
dard
ized
Res
idua
l
5.004.754.504.254.003.753.50
2
1
0
-1
-2
Residuals Versus the Fitted Values(response is logcost)