pertemuan 13 data deret waktu dan analisis regresi dan korelasi linier sederhana
DESCRIPTION
Mata kuliah : A0392 - Statistik Ekonomi Tahun: 2010. Pertemuan 13 Data Deret Waktu dan Analisis Regresi dan Korelasi Linier Sederhana. Outline Materi : Data Deret Waktu (Times Series) Analisis Regresi Linier Sederhana Koefisien Korelasi dan Uji Ketergantungan antar Peubah Acak. - PowerPoint PPT PresentationTRANSCRIPT
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Pertemuan 13Data Deret Waktu dan Analisis
Regresi dan Korelasi Linier Sederhana
Mata kuliah : A0392 - Statistik Ekonomi
Tahun : 2010
2
Outline Materi :
Data Deret Waktu (Times Series) Analisis Regresi Linier Sederhana Koefisien Korelasi dan Uji
Ketergantungan antar Peubah Acak
33
• Data deret berkala adalah sekumpulan data yang dicatat dalam suatu periode tertentu.
• Manfaat analisis data berkala adalah mengetahui kondisi masa mendatang.
• Peramalan kondisi mendatang bermanfaat untuk perencanaan produksi, pemasaran, keuangan dan bidang lainnya.
PENDAHULUAN
KOMPONEN DATA BERKALA
Trend; Variasi Musim; Variasi Siklus; dan Variasi yang Tidak Tetap (Irregular)
44
TREND
Suatu gerakan kecenderungan naik atau turun dalam jangka panjang yang diperoleh dari rata-rata perubahan dari waktu ke waktu dan nilainya cukup rata (smooth).
Tahun (X) Tahun (X)
Y Y
Trend Positif Trend Negatif
55
Metode Kuadrat Terkecil Untuk Trend Linier
Trend Pelanggan PT. Telkom
012345678
97 98 99 00 01
Tahun
Pe
lan
gg
an
(Ju
taa
n)
Data Y' Data Y
Y = a + bX
a = Y/N
b = YX/X2
Menentukan garis trend yang mempunyai jumlah terkecil dari kuadrat selisih data asli dengan data pada garis trendnya.
66
CONTOH METODE KUADRAT TERKECIL
Tahun Pelanggan =Y
Kode X(tahun)
Y.X X2
1997 5,0 -2 -10,0 4
1998 5,6 -1 -5,6 1
1999 6,1 0 0 0
2000 6,7 1 6,7 1
2001 7,2 2 14,4 4
Y=30,6 Y.X=5,5 X2=10
Nilai a = 30,6/5=6,12Nilai b =5,5/10=0,55Jadi persamaan trend Y’=6,12+0,55x
77
Y=a+bX+cX2
Y = a + bX + cX2
Koefisien a, b, dan c dicari dengan rumus sebagai berikut: a = (Y) (X4) – (X2Y) (X2)/ n (X4) - (X2)2
b = XY/X2
c = n(X2Y) – (X2 ) ( Y)/ n (X4) - (X2)2
Trend Kuadratis
0.002.00
4.006.00
8.00
97 98 99 00 01
TahunJu
mla
h P
ela
ng
ga
n
(ju
taa
n)
Untuk jangka waktu pendek, kemungkinan trend tidak bersifat linear. Metode kuadratis adalah contoh metode nonlinear
ANALISIS TREND KUADRATIS
88
CONTOH TREND KUADRATIS
Tahun Y X XY X2 X2Y X4
1997 5,0 -2 -10,00 4,00 20,00 16,00
1998 5,6 -1 -5,60 1,00 5,60 1,00
1999 6,1 0 0,00 0,00 0,00 0,00
2000 6,7 1 6,70 1,00 6,70 1,00
2001 7,2 2 14,40 4,00 2880 16,00
30.60
5,50 10,00 61,10 34,00
a = (Y) (X4) – (X2Y) (X2) = {(30,6)(34)-(61,1)(10)}/{(5)(34)-(10)2}=6,13 n (X4) - (X2)2
b = XY/X2 = 5,5/10=0,55c = n(X2Y) – (X2 ) ( Y) = {(5)(61,1)-(10)(30,6)}/{(5)(34)-(10)2}=-0,0071 n (X4) - (X2)2
Jadi persamaan kuadratisnya adalah Y =6,13+0,55x-0,0071x2
99
Y= a(1+b)X
Persamaan eksponensial dinyatakan dalam bentuk variabel waktu (X) dinyatakan sebagai pangkat. Untuk mencari nilai a, dan b dari data Y dan X, digunakan rumus sebagai berikut: Y’ = a (1 + b)X
Ln Y’ = Ln a + X Ln (1+b)Sehingga a = anti ln (LnY)/n b = anti ln (X. LnY) - 1 (X)2
Trend Eskponensial
0,00
5,00
10,00
15,00
97 98 99 00 01
Tahun
Jum
lah
Pel
angg
an
(juta
an)
ANALISIS TREND EKSPONENSIAL
1010
CONTOH TREND EKSPONENSIAL
Tahun Y X Ln Y X2 X Ln Y
1997 5,0 -2 1,6 4,00 -3,2
1998 5,6 -1 1,7 1,00 -1,7
1999 6,1 0 1,8 0,00 0,0
2000 6,7 1 1,9 1,00 1,9
2001 7,2 2 2,0 4,00 3,9
9,0 10,00 0,9
Nilai a dan b didapat dengan:a = anti ln (LnY)/n = anti ln 9/5=6,049b = anti ln (X. LnY) - 1 = {anti ln0,9/10}-1=0,094
(X)2 Sehingga persamaan eksponensial Y =6,049(1+0,094)x
1111
VARIASI MUSIM
Variasi musim terkait dengan perubahan atau fluktuasi dalam musim-musim atau bulan tertentu dalam 1 tahun.
Produksi Padi Permusim
0
10
20
30
I-
98
II-
98
III-
98
I-
99
II-
99
III-
99
I-
00
II-
00
III-
00
I-
01
II-
01
III-
03
Triw ulan
Prod
uksi
(000 t
on)
Pergerakan Inflasi 2002
0
0,5
1
1,5
2
2,5
1 2 3 4 5 6 7 8 9 10 11 12
Bulan
Infla
si (%
)
Indeks Saham PT. Astra Agro
Lestari, Maret 2003
0
50
100
150
03 05 13 14 22
Tanggal
Inde
ks
Variasi Musim Produk Pertanian
Variasi Inflasi Bulanan
Variasi Harga Saham Harian
1212
VARIASI MUSIM DENGAN METODE RATA-RATA SEDERHANA
Indeks Musim = (Rata-rata per kuartal/rata-rata total) x 100Bulan Pendapatan Rumus= Nilai bulan ini x
100 Nilai rata-rata
Indeks Musim
Januari 88 (88/95) x100 93
Februari 82 (82/95) x100 86
Maret 106 (106/95) x100 112
April 98 (98/95) x100 103
Mei 112 (112/95) x100 118
Juni 92 (92/95) x100 97
Juli 102 (102/95) x100 107
Agustus 96 (96/95) x100 101
September 105 (105/95) x100 111
Oktober 85 (85/95) x100 89
November 102 (102/95) x100 107
Desember 76 (76/95) x100 80
Rata-rata 95
1313
METODE RATA-RATA DENGAN TREND
• Metode rata-rata dengan trend dilakukan dengan cara yaitu indeks musim diperoleh dari perbandingan antara nilai data asli dibagi dengan nilai trend.
• Oleh sebab itu nilai trend Y’ harus diketahui dengan persamaan Y’ = a + bX.
1414
METODE RATA-RATA DENGAN TREND
Bulan Y Y’ Perhitungan Indeks Musim
Januari 88 97,41 (88/97,41) x 100 90,3
Februari 82 97,09 (82/97,09) x 100 84,5
Maret 106 96,77 (106/96,77) x100 109,5
April 98 96,13 (98/96,13) x 100 101,9
Mei 112 95,81 (112/95,81) x 100 116,9
Juni 92 95,49 (92/95,49) x 100 96,3
Juli 102 95,17 (102/95,17) x 100 107,2
Agustus 96 94,85 (96/94,85) x 100 101,2
September 105 94,53 (105/94,53) x 100 111,1
Oktober 85 93,89 (85/93,89) x 100 90,5
November 102 93,57 (102/93,57) x 100 109,0
Desember 76 93,25 (76/93,25) x 100 81,5
1515
VARIASI SIKLUS
Siklus
Ingat
Y = T x S x C x I
MakaTCI = Y/SCI = TCI/T
Di mana CI adalah Indeks Siklus
Siklus Indeks Saham Gabungan
-2,5
-2
-1,5
-1
-0,5
0
0,5
1
1,5
2
2,5
94 95 96 97 98 99 00 01 02
Tahun
IHS
G
1616
CONTOH SIKLUS
Th Trwl Y T S TCI=Y/S CI=TCI/T C
I 22 17,5
1998 II 14 17,2 95 14,7 86
III 8 16,8 51 15,7 93 92
I 25 16,5 156 16,0 97 97
1999 II 15 16,1 94 16,0 99 100
III 8 15,8 49 16,3 103 102
I 26 15,4 163 16,0 104 104
2000 II 14 15,1 88 15,9 105 105
III 8 14,7 52 15,4 105 106
I 24 14,3 157 15,3 107 108
2001 II 14 14,0 89 15,7 112
III 9 13,6
1717
GERAK TAK BERATURAN
SiklusIngat Y = T x S x C x ITCI = Y/SCI = TCI/TI = CI/C
Perkembangan Inflasi dan Suku Bunga
-10
0
10
20
30
40
50
60
70
80
94 95 96 97 98 99 00 01 02
Tahun
Inflasi Suku Bunga
1818
Th Trwl CI=TCI/T C I=(CI/C) x 100
I
1998 II
86
III
93 92 101
I
97 97 100
1999 II
99 100 99
III
103 102 101
I
104 104 100
2000 II
105 105 100
III
105 106 99
I
107 108 99
2001 II
112
III
GERAK TAK BERATURAN
19
PENGUJIAN KOEFISIEN REGRESI DENGAN
ANALISIS VARIANSI
20
Measures of Variation: The Sum of Squares
SST = SSR + SSE
Total Sample
Variability
= Explained Variability
+ Unexplained Variability
SST = Total Sum of Squares
SSR = Regression Sum of Squares
SSE = Error Sum of Squares
21
Measures of Variation: The Sum of Squares
Xi
Y
X
Y
SST = (Yi - Y)2
SSE =(Yi - Yi )2
SSR = (Yi - Y)2
_
_
_
22
Venn Diagrams and Explanatory Power of Regression
Sales
Sizes
Variations in Sales explained by Sizes or variations in Sizes used in explaining variation in Sales
Variations in Sales explained by the error term or unexplained by Sizes
Variations in store Sizes not used in explaining variation in Sales
SSE
SSR
23
The ANOVA Table in Excel
ANOVA
df SS MS FSignificance
F
Regression
kSSR
MSR
=SSR/kMSR/MSE
P-value of
the F Test
Residualsn-k-1
SSE
MSE
=SSE/(n-k-1)
Total n-1SST
24
Measures of VariationThe Sum of Squares: Example
ANOVA
df SS MS F Significance F
Regression 1 30380456.12 30380456 81.17909 0.000281201
Residual 5 1871199.595 374239.92
Total 6 32251655.71
Excel Output for Produce Stores
SSR
SSERegression (explained) df
Degrees of freedom
Error (residual) df
Total df
SST
25
Venn Diagrams and Explanatory Power of Regression
Sales
Sizes
2
SSR
SSR S
r
SE
26
Standard Error of Estimate
•
• Measures the standard deviation (variation) of the Y values around the regression equation
2
1
ˆ
2 2
n
ii
YX
Y YSSE
Sn n
27
Measures of Variation: Produce Store Example
Regression StatisticsMultiple R 0.9705572R Square 0.94198129Adjusted R Square 0.93037754Standard Error 611.751517Observations 7
Excel Output for Produce Stores
r2 = .94
94% of the variation in annual sales can be explained by the variability in the size of the store as measured by square footage.
Syxn
28
Linear Regression Assumptions
• Normality– Y values are normally distributed for each X– Probability distribution of error is normal
• Homoscedasticity (Constant Variance)
• Independence of Errors
29
Consequences of Violationof the Assumptions
• Violation of the Assumptions– Non-normality (error not normally distributed)– Heteroscedasticity (variance not constant)
• Usually happens in cross-sectional data– Autocorrelation (errors are not independent)
• Usually happens in time-series data• Consequences of Any Violation of the Assumptions
– Predictions and estimations obtained from the sample regression line will not be accurate
– Hypothesis testing results will not be reliable• It is Important to Verify the Assumptions
30
• Y values are normally distributed around the regression line.
• For each X value, the “spread” or variance around the regression line is the same.
Variation of Errors Aroundthe Regression Line
X1
X2
X
Y
f(e)
Sample Regression Line
31
Inference about the Slope: t Test
• t Test for a Population Slope– Is there a linear dependency of Y on X ?
• Null and Alternative Hypotheses– H0: 1 = 0 (no linear dependency)
– H1: 1 0 (linear dependency)
• Test Statistic–
–
1
1
1 1
2
1
where
( )
YXb n
bi
i
b St S
SX X
. . 2d f n
32
Example: Produce Store
Data for 7 Stores:Estimated Regression Equation:Annual
Store Square Sales Feet ($000)
1 1,726 3,681
2 1,542 3,395
3 2,816 6,653
4 5,555 9,543
5 1,292 3,318
6 2,208 5,563
7 1,313 3,760
ˆ 1636.415 1.487i iY X
The slope of this model is 1.487.
Does square footage affect annual sales?
33
Inferences about the Slope: t Test Example
H0: 1 = 0
H1: 1 0
.05
df 7 - 2 = 5
Critical Value(s):
Test Statistic:
Decision:
Conclusion:There is evidence that square footage affects annual sales.
t0 2.5706-2.5706
.025
Reject Reject
.025
From Excel Printout
Reject H0.
Coefficients Standard Error t Stat P-valueIntercept 1636.4147 451.4953 3.6244 0.01515Footage 1.4866 0.1650 9.0099 0.00028
1b 1bS t
p-value
34
Inferences about the Slope: Confidence Interval Example
Confidence Interval Estimate of the Slope:
11 2n bb t S Excel Printout for Produce Stores
At 95% level of confidence, the confidence interval for the slope is (1.062, 1.911). Does not include 0.
Conclusion: There is a significant linear dependency of annual sales on the size of the store.
Lower 95% Upper 95%Intercept 475.810926 2797.01853Footage 1.06249037 1.91077694
35
Inferences about the Slope: F Test
• F Test for a Population Slope– Is there a linear dependency of Y on X ?
• Null and Alternative Hypotheses– H0: 1 = 0 (no linear dependency)
– H1: 1 0 (linear dependency)
• Test Statistic
–
– Numerator d.f.=1, denominator d.f.=n-2
1
2
SSR
FSSEn
36
Relationship between a t Test and an F Test
• Null and Alternative Hypotheses– H0: 1 = 0 (no linear dependency)– H1: 1 0 (linear dependency)
•
• The p –value of a t Test and the p –value of an F Test are Exactly the Same
• The Rejection Region of an F Test is Always in the Upper Tail
2
2 1, 2n nt F
37
ANOVAdf SS MS F Significance F
Regression 1 30380456.12 30380456.12 81.179 0.000281Residual 5 1871199.595 374239.919Total 6 32251655.71
Inferences about the Slope: F Test Example
Test Statistic:
Decision:Conclusion:
H0: 1 = 0H1: 1 0 .05numerator df = 1denominator df 7 - 2 = 5
There is evidence that square footage affects annual sales.
From Excel Printout
Reject H0.
0 6.61
Reject
= .05
1, 2nF
p-value
38
Purpose of Correlation Analysis
• Correlation Analysis is Used to Measure Strength of Association (Linear Relationship) Between 2 Numerical Variables– Only strength of the relationship is concerned– No causal effect is implied
39
Purpose of Correlation Analysis
• Population Correlation Coefficient (Rho) is Used to Measure the Strength between the Variables
XY
X Y
40
• Sample Correlation Coefficient r is an Estimate of and is Used to Measure the Strength of the Linear Relationship in the Sample Observations
Purpose of Correlation Analysis
(continued)
1
2 2
1 1
n
i ii
n n
i ii i
X X Y Yr
X X Y Y
41r = .6 r = 1
Sample Observations from Various r Values
Y
X
Y
X
Y
X
Y
X
Y
X
r = -1 r = -.6 r = 0
42
Features of and r
• Unit Free
• Range between -1 and 1
• The Closer to -1, the Stronger the Negative Linear Relationship
• The Closer to 1, the Stronger the Positive Linear Relationship
• The Closer to 0, the Weaker the Linear Relationship
43
• Hypotheses – H0: = 0 (no correlation)
– H1: 0 (correlation)
• Test Statistic
–
2
2 1
2 2
1 1
where
2n
i ii
n n
i ii i
rt
rn
X X Y Yr r
X X Y Y
t Test for Correlation
44
Example: Produce Stores
Regression StatisticsMultiple R 0.9705572R Square 0.94198129Adjusted R Square 0.93037754Standard Error 611.751517Observations 7
From Excel Printout r
Is there any evidence of linear relationship between annual sales of a store and its square footage at .05 level of significance?
H0: = 0 (no association)
H1: 0 (association)
.05
df 7 - 2 = 5
45
Example: Produce Stores Solution
0 2.5706-2.5706
.025
Reject Reject
.025
Critical Value(s):
Conclusion:There is evidence of a linear relationship at 5% level of significance.
Decision:Reject H0.
2
.97069.0099
1 .942052
rt
rn
The value of the t statistic is exactly the same as the t statistic value for test on the slope coefficient.
46
Estimation of Mean Values
Confidence Interval Estimate for :
The Mean of Y Given a Particular Xi
2
22
1
( )1ˆ
( )
ii n YX n
ii
X XY t S
n X X
t value from table with df=n-2
Standard error of the estimate
Size of interval varies according to distance away from mean, X
| iY X X
47
Prediction of Individual Values
Prediction Interval for Individual Response Yi at a Particular Xi
Addition of 1 increases width of interval from that for the mean of Y
2
22
1
( )1ˆ 1( )
ii n YX n
ii
X XY t S
n X X
48
Interval Estimates for Different Values of X
Y
X
Prediction Interval for a Individual Yi
a given X
Confidence Interval for the Mean of Y
Y i = b0 + b1X i
X
49
Example: Produce Stores
Yi = 1636.415 +1.487Xi
Data for 7 Stores:
Regression Model Obtained:
Annual Store Square Sales
Feet ($000)
1 1,726 3,681
2 1,542 3,395
3 2,816 6,653
4 5,555 9,543
5 1,292 3,318
6 2,208 5,563
7 1,313 3,760
Consider a store with 2000 square feet.
50
Estimation of Mean Values: Example
Find the 95% confidence interval for the average annual sales for stores of 2,000 square feet.
2
22
1
( )1ˆ 4610.45 612.66( )
ii n YX n
ii
X XY t S
n X X
Predicted Sales Yi = 1636.415 +1.487Xi = 4610.45 ($000)
X = 2350.29 SYX = 611.75 tn-2 = t5 = 2.5706
Confidence Interval Estimate for| iY X X
|3997.02 5222.34iY X X
51
Prediction Interval for Y : Example
Find the 95% prediction interval for annual sales of one particular store of 2,000 square feet.
Predicted Sales Yi = 1636.415 +1.487Xi = 4610.45 ($000)
X = 2350.29 SYX = 611.75 tn-2 = t5 = 2.5706
2
22
1
( )1ˆ 1 4610.45 1687.68( )
ii n YX n
ii
X XY t S
n X X
Prediction Interval for Individual
2922.00 6297.37iX XY
iX XY
5252
PENGGUNAAN MS EXCEL UNTUK REGRESI
• Masukkan data Y dan data X pada sheet MS Excel, misalnya data Y di kolom A dan X pada kolom B dari baris 1 sampai 5.
• Klik icon tools, pilih ‘data analysis’, dan pilih ‘simple linear regression’.
• Pada kotak data tertulis Y variable cell range: masukkan data Y dengan mem-blok kolom a atau a1:a5. Pada X variable cell range: masukkan data X dengan mem-blok kolom b atau b1:b5.
• Anda klik OK, maka hasilnya akan keluar. Y’= a+b X; a dinyatakan sebagai intercept dan b sebagai X variable1 pada kolom coefficients.
5353
5454
5555
5656
SELAMAT BELAJAR SEMOGA SUKSES SELALU