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Pertemuan 13 Data Deret Waktu dan Analisis Regresi dan Korelasi Linier Sederhana

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.

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Pertemuan 13 Data Deret Waktu dan Analisis Regresi dan Korelasi Linier Sederhana

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  1. Mata kuliah : A0392 - Statistik Ekonomi Tahun : 2010 Pertemuan 13Data Deret Waktu dan Analisis Regresi dan KorelasiLinier Sederhana

  2. Outline Materi : • Data Deret Waktu (Times Series) • Analisis Regresi Linier Sederhana • Koefisien Korelasi dan Uji Ketergantungan antar Peubah Acak

  3. 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

  4. 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

  5. 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

  6. 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 CONTOH METODE KUADRAT TERKECIL Nilai a = 30,6/5=6,12 Nilai b =5,5/10=0,55 Jadi persamaan trend Y’=6,12+0,55x 6

  7. 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

  8. 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 CONTOH TREND KUADRATIS 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

  9. 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

  10. 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=6,049 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 CONTOH TREND EKSPONENSIAL 10

  11. 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

  12. 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 (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 VARIASI MUSIM DENGAN METODE RATA-RATA SEDERHANA Indeks Musim = (Rata-rata per kuartal/rata-rata total) x 100 12

  13. 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

  14. a.Menghitung indeks musim = (nilai data asli/nilai trend) x 100 METODE RATA-RATA DENGAN TREND 14

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

  16. 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 CONTOH SIKLUS 16

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

  18. 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 18

  19. PENGUJIAN KOEFISIEN REGRESI DENGAN ANALISIS VARIANSI

  20. Measures of Variation: The Sum of Squares SST = SSR + SSE Total Sample Variability UnexplainedVariability = ExplainedVariability + SST = Total Sum of Squares SSR = Regression Sum of Squares SSE = Error Sum of Squares

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

  22. 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

  23. The ANOVA Table in Excel

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

  25. Venn Diagrams and Explanatory Power of Regression Sales Sizes

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

  27. 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.

  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 (errornot 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. Variation of Errors Aroundthe 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

  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

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

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

  34. 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.

  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

  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

  37. 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 There is evidence that square footage affects annual sales.  = .05 0 6.61

  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

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

  41. 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

  42. 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

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

  44. Example: Produce Stores r 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

  45. 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 0 2.5706

  46. 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

  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

  48. 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

  49. 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

  50. 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

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