Describing Associations, Covariance, Correlation, and Causality. Interest Rates and Inflation. Data & Scatter Diagram. Lecture 4

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1 Describing Associations, Covariance, Correlation, and Causality Lecture Reading: Chapter 6 & SW11 ( Readings in portal) 1 Interest Rates and Inflation At the heart of Canada s monetary policy framework is the inflation-control target. The target for inflation is the per cent midpoint of a control range of 1 to 3 per cent. Inflation is measured as the year-over-year rate of increase in the total consumer price inde (CPI). The Bank also monitors a set of core inflation measures, including the CPIX which strips out eight of the most volatile CPI components. The Bank carries out monetary policy through changes in its policy interest rate the Target for the Overnight Rate. Monetary policy actions (changes in the policy rate) take time usually between si and eight quarters to work their way through the economy and to have their full effect on inflation. date Inflation rate (CPIX) Data & Scatter Diagram Monthly Data: Sep -Jan 13 (n=11) Interest Rate (Target Overnight Rate 1 months earlier) Which kind of data are these? 3 Lecture, Page 1 of 9

2 Qualitative Assessments Use scatter diagram to qualitatively assess relationship between two variables: positive linear relationship negative linear relationship non-linear relationship no relationship strong relationship weak relationship 1 1 y y6 y y8 1 (A) No relationship (B) Linear relationship (C) A constant relationship (D) Weak horizontal relationship (E) -Strong horizontal 1 3 relationship What qualitative assessments should we make? Qualitative then Quantitative Why scattered? Qualitative ass.? Statistics quantify strength Strength depends on scatter & slope For linear cases: covariance, correlation, R, OLS (slope) Inflation rate (CPIX) Monthly Data: Sep -Jan 13 (n=11) Interest Rate (Target Overnight Rate 1 months earlier) 6 Lecture, Page of 9

3 Covariance Covariance: How two variables vary with respect to each other co: with, together, joint variance: vary about mean σ y = N i=1 ( i μ X )(y i μ Y ) s y = N n i=1 ( i X )(y i Y ) n 1 Units of measurement? Zero covariance: no linear relationship Positive covariance: when X big Y tends to be big; when X small Y tends to be small Negative covariance: when X is big Y tends to be small & v.v. 7 Formula and Intuition Female ER (%) s y = 1 Employment Rates n = 3 OECD countries covariance = Male ER (%) n i=1 but, Meico ( i X )(y i Y ) n 1 Approimate mean of X? Y? If i and y i are both above average (e.g. Iceland with ER of 78% for females & 81% for males)? If i and y i are both below average (e.g. Greece with ER of % for females & 61% for males)? 8 ER (%) , n = 3 OECD covariance = Population (1,'s) ER (%) , n = 3 OECD covariance = Population (1,,'s) Female ER (%) 8 6 1, n = 3 OECD covariance = Male ER (%) 8 Which is strongest relationship? Covariance can only indicate the direction of a linear relationship: nothing about strength 9 Lecture, Page 3 of 9

4 Coefficient of Correlation Parameter rho : ρ = σ y σ σ y Statistic: r = s y s s y Measures strength of a linear relationship between two variables: values from -1 to 1 What is σ y? What is s? What about sign? What are the units of measurement? Value near -1 strong neg. linear relationship Value near 1 strong pos. linear relationship Value near no linear relationship 1 Are the two measures of inflation the official CPI and the online measure positively or negatively correlated? The Billion Prices Project: Using Online Prices for Measurement and Research, Spring 16, Journal of Economic Perspectives =1.17/jep ER (%) , n = 3 OECD correlation = Population (1,'s) ER (%) , n = 3 OECD correlation = Population (1,,'s) Female ER (%) 8 6 1, n = 3 OECD correlation = Male ER (%) 8 Which is strongest relationship? 1 Lecture, Page of 9

5 y1 y3 corr(,y1) = corr(,y3) = y y corr(,y) = corr(,y) = Which concept do these graphs illustrate? 13 y1 y3 corr(,y1) = corr(,y3) = y y corr(,y) = corr(,y) = Which concept do these graphs illustrate? 1 Correlation: Use Only to Measure Strength of Linear Relationships Obtain nonsense if use on non-linear relationships No relationship versus no linear relationship: not the same thing Association versus Correlation : not the same thing y 3 1 n = 1 r =.11, cov = Lecture, Page of 9

6 Linking Lecture & Tetbook CPIX n = 11 months r =.39, cov = Lagged Target Rate Standardized CPIX 3 n = 11 months r =.39, cov = Standardized Lag TR Once you standardize, the covariance is mathematically equal to the correlation. 16 correlate debt_pct_gdp_9 emp_rate_9 fem_emp_rate_9 male_emp_rate_9; Correlation Matri (obs=3) deb~9 emp~9 fem~9 mal~ debt_pc~9 1. emp_rat~ fem_emp~ male_em~ correlate debt_pct_gdp_9 emp_rate_9 fem_emp_rate_9 male_emp_rate_9, covariance; Variance-Covariance Matri (obs=3) deb~9 emp~9 fem~9 mal~ debt_pc~ emp_rat~ fem_emp~ male_em~ What is s.d. of debt %? Source: OECD website, n = 3 OECD members 17 correlate debt_pct_gdp_9 emp_rate_9 fem_emp_rate_9 male_emp_rate_9 if country~="japan" (obs=33) deb~9 emp~9 fem~9 mal~ debt_pc~9 1. emp_rat~ fem_emp~ male_em~ correlate debt_pct_gdp_9 emp_rate_9 fem_emp_rate_9 male_emp_rate_9 if country~="japan", covariance; (obs=33) deb~9 emp~9 fem~9 mal~ debt_pc~9 86. emp_rat~ fem_emp~ male_em~ Source: OECD website, n = 3 OECD members Why is 86. so much smaller? 18 Lecture, Page 6 of 9

7 Research Question Research question: Inquires about the causal relationship among variables Eample: What is the effect of lecture attendance on learning? Attendance Learning X variable ( eplanatory ): Attendance Y variable ( dependent ): Learning How big is the effect? (skinny or thick arrow?) 19 Observational Data Observational Data: What has actually happened to agents (people, countries, firms, etc) where all variables are likely affected by choices/behaviors of agents and unobserved variables that affect both the dependent and independent variable Unobserved variables: Not in your data and affect both your and y variable aka lurking, confounding, or omitted variables Observational Data Attendance Learning Motivation, effort, quality of classes, 1 Lecture, Page 7 of 9

8 Eperimental Data Eperimental Data: Data collected in an eperimental setting where the values of the variable are set by the researcher Researchers usually randomly set values for eplanatory variable(s) and see reactions Dosage (mg) Sleep (hours) Patient Characteristics What is key difference from observational data? Eperimental Data: Drug Trial Dosage Hours of i (mg) i Sleep y i X-bar =.61 Y-bar =.66 s = 1.1 s y = 1.18 s y = 1.9 Hours of Sleep Dosage (mg) How do patient characteristics factor into this graph? Can we infer causality? 3 Eogenous, Endogenous & Endogeneity Bias Eogenous: X variable not associated with factors that also affect Y (no lurking variables) Endogenous: X variable is associated with factors that also affect Y (lurking variables) Endogeneity bias: Observed correlation driven by unobserved variables Correlation either overstates or understates any truly causal relationship between the variables Spurious correlation: false/misleading correlation Lecture, Page 8 of 9

Leveling Up: Early Results from a Randomized Evaluation of Post-Secondary Aid Abstract: Does financial aid increase college attendance and completion?

9 Leveling Up: Early Results from a Randomized Evaluation of Post-Secondary Aid Abstract: Does financial aid increase college attendance and completion? Selection bias and the high implicit ta rates imposed by overlapping aid programs make this question difficult to answer. This paper reports initial findings from a randomized evaluation of a large privately-funded scholarship program for applicants to Nebraska s public colleges and universities. Our research design answers the challenges of aid evaluation with random assignment of aid offers and a strong first stage for aid received: randomly assigned aid offers increased aid received markedly. This in turn appears to have boosted enrollment and persistence, while also shifting many applicants from two- to four-year schools. cont d net slide Research question? Observational or eperimental data? Bias? Source: Angrist et al (1) Note: To access NBER papers, use a U of T computer. Leveling Up: Early Results from a Randomized Evaluation of Post-Secondary Aid Abstract cont d: Awards offered to nonwhite applicants, to those with relatively low academic achievement, and to applicants who targeted lessselective four-year programs (as measured by admissions rates) generated the largest gains in enrollment and persistence, while effects were much smaller for applicants predicted to have stronger post-secondary outcomes in the absence of treatment. Thus, awards enabled groups with historically-low college attendance to level up, largely equalizing enrollment and persistence rates with traditionally college-bound peers, particularly at four-year programs. Awards offered to prospective community college students had little effect on college enrollment or the type of college attended. This part discusses interaction effects, which we will study in Chapter 1/Lecture (part of multiple regression analysis) 6 7 Lecture, Page 9 of 9

Controlling for Time Invariant Heterogeneity

Controlling for Time Invariant Heterogeneity Yona Rubinstein July 2016 Yona Rubinstein (LSE) Controlling for Time Invariant Heterogeneity 07/16 1 / 19 Observables and Unobservables Confounding Factors