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

KOMPONEN DATA BERKALA PENDAHULUAN 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. KOMPONEN DATA BERKALA Trend; Variasi Musim; Variasi Siklus; dan Variasi yang Tidak Tetap (Irregular) 3

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). Y Y Tahun (X) Tahun (X) Trend Positif Trend Negatif 4

Metode Kuadrat Terkecil Untuk Trend Linier Menentukan garis trend yang mempunyai jumlah terkecil dari kuadrat selisih data asli dengan data pada garis trendnya. Y = a + bX a = Y/N b = YX/X2 5

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 2000 6,7 2001 7,2 2 14,4   Y=30,6 Y.X=5,5 X2=10 Nilai a = 30,6/5=6,12 Nilai b =5,5/10=0,55 Jadi persamaan trend Y’=6,12+0,55x 6

ANALISIS TREND KUADRATIS Untuk jangka waktu pendek, kemungkinan trend tidak bersifat linear. Metode kuadratis adalah contoh metode nonlinear 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 7

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 1999 6,1 0,00 2000 6,7 1 6,70 2001 7,2 2 14,40 2880   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=0,55 c = 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 8

ANALISIS TREND EKSPONENSIAL 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 Y= a(1+b)X 9

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 1,8 0,00 0,0 2000 6,7 1 1,9 2001 7,2 2 2,0 3,9   9,0 10,00 0,9 Nilai a dan b didapat dengan: a = anti ln (LnY)/n = anti ln 9/5=6,049 b = anti ln  (X. LnY) - 1 = {anti ln0,9/10}-1=0,094 (X)2 Sehingga persamaan eksponensial Y =6,049(1+0,094)x 10

VARIASI MUSIM Variasi musim terkait dengan perubahan atau fluktuasi dalam musim-musim atau bulan tertentu dalam 1 tahun. Variasi Musim Produk Pertanian Variasi Harga Saham Harian Variasi Inflasi Bulanan 11

VARIASI MUSIM DENGAN METODE RATA-RATA SEDERHANA Indeks Musim = (Rata-rata per kuartal/rata-rata total) x 100 Bulan 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 (112/95) 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 Desember 76 (76/95) x100 80 Rata-rata 95   12

METODE RATA-RATA DENGAN TREND a.      Menghitung indeks musim = (nilai data asli/nilai trend) x 100   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.   13

METODE RATA-RATA DENGAN TREND a.      Menghitung indeks musim = (nilai data asli/nilai trend) x 100   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 93,57 (102/93,57) x 100 109,0 Desember 76 93,25 (76/93,25) x 100 81,5   14

VARIASI SIKLUS TCI = Y/S CI = TCI/T Siklus Ingat Y = T x S x C x I Maka TCI = Y/S CI = TCI/T Di mana CI adalah Indeks Siklus 15

CONTOH SIKLUS Th Trwl Y T S TCI=Y/S CI=TCI/T C I 22 17,5 1998 II 14   I 22 17,5 1998 II 14 17,2 95 14,7 86 III 8 16,8 51 15,7 93 92 25 16,5 156 16,0 97 1999 15 16,1 94 99 100 15,8 49 16,3 103 102 26 15,4 163 104 2000 15,1 88 15,9 105 52 106 24 14,3 157 15,3 107 108 2001 14,0 89 112 9 13,6 16

GERAK TAK BERATURAN Siklus Ingat Y = T x S x C x I TCI = Y/S CI = TCI/T I = CI/C 17

GERAK TAK BERATURAN Th Trwl CI=TCI/T C I=(CI/C) x 100 I 1998 II 86 III   I 1998 II 86 III 93 92 101 97 100 1999 99 103 102 104 2000 105 106 107 108 2001 112 18

PENGUJIAN KOEFISIEN REGRESI DENGAN ANALISIS VARIANSI

Measures of Variation: The Sum of Squares SST = SSR + SSE Total Sample Variability = Unexplained Variability Explained Variability + SST = Total Sum of Squares SSR = Regression Sum of Squares SSE = Error Sum of Squares

Measures of Variation: The Sum of Squares Y  SSE =(Yi - Yi )2 _ SST = (Yi - Y)2 _  SSR = (Yi - Y)2 _ Y X Xi

Venn Diagrams and Explanatory Power of Regression Variations in Sales explained by the error term or unexplained by Sizes Variations in store Sizes not used in explaining variation in Sales Sales Variations in Sales explained by Sizes or variations in Sizes used in explaining variation in Sales Sizes

The ANOVA Table in Excel df SS MS F Significance Regression k SSR MSR =SSR/k MSR/MSE P-value of the F Test Residuals n-k-1 SSE MSE =SSE/(n-k-1) Total n-1 SST

Measures of Variation The Sum of Squares: Example Excel Output for Produce Stores Degrees of freedom Regression (explained) df SST SSE Error (residual) df SSR Total df

Venn Diagrams and Explanatory Power of Regression Sales Sizes

Standard Error of Estimate Measures the standard deviation (variation) of the Y values around the regression equation

Measures of Variation: Produce Store Example Excel Output for Produce Stores n Syx 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.

Linear Regression Assumptions Normality Y values are normally distributed for each X Probability distribution of error is normal Homoscedasticity (Constant Variance) Independence of Errors

Consequences of 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

Variation of Errors Around the Regression Line Y values are normally distributed around the regression line. For each X value, the “spread” or variance around the regression line is the same. f(e) Y X2 X1 X Sample Regression Line

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

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 The slope of this model is 1.487. Does square footage affect annual sales?

Inferences about the Slope: t Test Example Test Statistic: Decision: Conclusion: H0: 1 = 0 H1: 1  0   .05 df  7 - 2 = 5 Critical Value(s): From Excel Printout Reject H0. Reject Reject p-value .025 .025 There is evidence that square footage affects annual sales. t -2.5706 2.5706

Inferences about the Slope: Confidence Interval Example Confidence Interval Estimate of the Slope: 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.

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

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

Inferences about the Slope: F Test Example Test Statistic: Decision: Conclusion: H0: 1 = 0 H1: 1  0   .05 numerator df = 1 denominator df  7 - 2 = 5 From Excel Printout p-value Reject H0. Reject  = .05 There is evidence that square footage affects annual sales. 6.61

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

Purpose of Correlation Analysis Population Correlation Coefficient  (Rho) is Used to Measure the Strength between the Variables

Purpose of Correlation Analysis (continued) Sample Correlation Coefficient r is an Estimate of  and is Used to Measure the Strength of the Linear Relationship in the Sample Observations

Sample Observations from Various r Values Y Y Y X X X r = -1 r = -.6 r = 0 Y Y X X r = .6 r = 1

Features of r 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

t Test for Correlation Hypotheses Test Statistic H0:  = 0 (no correlation) H1:   0 (correlation) Test Statistic

Example: Produce Stores From Excel Printout 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

Example: Produce Stores Solution Decision: Reject H0. Conclusion: There is evidence of a linear relationship at 5% level of significance. Critical Value(s): Reject Reject The value of the t statistic is exactly the same as the t statistic value for test on the slope coefficient. .025 .025 -2.5706 2.5706

Estimation of Mean Values Confidence Interval Estimate for : The Mean of Y Given a Particular Xi Size of interval varies according to distance away from mean, Standard error of the estimate t value from table with df=n-2

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

Interval Estimates for Different Values of X Confidence Interval for the Mean of Y Prediction Interval for a Individual Yi Y  Yi = b0 + b1Xi X a given X

Example: Produce Stores Data for 7 Stores: 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. Regression Model Obtained:  Yi = 1636.415 +1.487Xi

Estimation of Mean Values: Example Confidence Interval Estimate for Find the 95% confidence interval for the average annual sales for stores of 2,000 square feet.  Predicted Sales Yi = 1636.415 +1.487Xi = 4610.45 ($000) tn-2 = t5 = 2.5706 X = 2350.29 SYX = 611.75

Prediction Interval for Y : Example Prediction Interval for Individual 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) tn-2 = t5 = 2.5706 X = 2350.29 SYX = 611.75

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. 52

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SELAMAT BELAJAR SEMOGA SUKSES SELALU 56