SDG Data Structure Definition Overview

SDG Data Structure Definition (DSD) is a framework developed by the Working Group on SDMX for SDG Indicators, established by IAEG-SDGs in April 2016. It provides a standardized approach for organizing and reporting data related to Sustainable Development Goals (SDGs). The current version, 1.14, was released in July 2023 and includes guidelines for utilizing the Global SDG DSD and customizing it according to specific needs. The DSD encompasses dimensions such as Frequency, Reporting Type, Series, and Reference Area, each serving a distinct purpose in categorizing and analyzing SDG indicators.

• SDG Indicators
• Data Structure
• IAEG-SDGs
• Global DSD
• Reporting Type

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1. Chapter 3 Multiple Regression

2. Learning Objectives Derive the ordinary least-squares estimators of the regression coefficients when there are two or more right-hand variables in our model Fit a multiple regression model using the least- squares criterion Identify the conditions under which a multiple regression estimate is the same as the simple regression estimate Interpret a multiple regression coefficient

3. API and Free Lunch Eligibility (FLE) for 20 California Elementary Schools 1000 950 900 API 850 800 750 700 0 20 40 60 80 100 FLE (%) Low PE Mid PE High PE

4. API and Free Lunch Eligibility (FLE) for 20 California Elementary Schools 1000 y = -1.37x + 985 950 900 API 850 y = -0.71x + 879 800 750 y = 0.08x + 755 700 0 20 40 60 80 100 FLE (%) Low PE Mid PE High PE

5. Multiple Regression Model with K+1 Parameters = + + + + + ... Y b b X b X b X e 0 1 1 2 2 i i i K Ki i If we have two right-hand variables, our model looks like this: = i Y b + + + b X b X e 0 1 1 2 2 i i i Figure 3.1. Illustration of a regression plane for a model with two explanatory variables, X1 and X2

6. Deriving Least-Squares Estimator in Multiple Regression N N = = 2 i 2 min = ( ... ) SSE e Y b b X b X 0 1 1 i i K Ki , 0,1,..., b k K k = = 1 1 i i N i )( 1) = : 2( ... 0 b Y b b X b X 0 0 1 1 i i K Ki = 1 N i = : 2( ... )( ) 0 b Y b b X b X X 1 0 1 1 1 i i K Ki i = 1 N i = : 2( ... )( ) 0 b Y b b X b X X 2 0 1 1 2 i i K Ki i = 1 N i = : 2( ... )( ) 0 b Y b b X b X X 0 1 1 K i i K Ki Ki = 1

7. Two RHS Variables 3 Equations, 3 Unknowns (b0, b1, b2, after dividing by 2) N i )( 1) = : ( 0 b Y b b X b X 0 0 1 1 2 2 i i i = 1 N i = : ( )( ) 0 b Y b b X b X X 1 0 1 1 2 2 1 i i i i = 1 N i = : ( )( ) 0 b Y b b X b X X 2 0 1 1 2 2 2 i i i i = 1

8. OLS Formula, 2 RHS Variables = b Y b X b X 0 1 1 2 2 When do these collapse to the simple OLS formulas? N N N N i i i i 2 2 x y x x x x y 1 1 2 2 i i i i i i i = = = = = 1 1 1 1 b N = x y 1 2 N N N 1 i i i i 2 1 2 2 x x x x s = i 1 N i b 1 2 i i i i 1 = = = 2 1 1 1 1 i x i = 1 N N N N i i i i 2 1 x y x x x x y 2 1 2 1 i i i i i i i = = = = = 1 1 1 1 b 2 2 N N N i i 2 1 2 2 x x x x 1 2 i i i i = = = 1 1 1 i

9. Perfect Multicollinearity: Least squares fails if X1 and X2 are perfectly correlated (i.e., X1 = c*X2) N N N N i i i i 2 ( ) x y cx x cx cx y 1 1 1 1 1 i i i i i i i = = = = = 1 1 1 1 b 1 N N N i i 2 1 2 2 ( ) ( ) x cx x cx 1 1 1 i i i i = = = 1 1 i 1 i N N N N i i N 2 2 1 2 1 c x y x x x y 1 1 i i i i i i = = = = = 1 1 1 1 i N N i i 2 2 1 2 1 2 2 1 ) i x ( c x x i i = = = 1 1 1 i 0 0 =

10. 20 Elementary Schools 20 Elementary Schools We have two regression models = = + + 951.87 777.17 0.51 2.11 API API FLE FLE e i i i + 2.34 PE e i i i i Why is the coefficient on FLE smaller? Which model has the larger R2? Why?

11. Interpreting MR Coefficients Interpreting MR Coefficients = + + + Y b b X b X e 0 1 1 2 2 i i i i Suppose two individuals have the same X2 value and their X1 values differ by one unit. The predicted difference in their Y values is b1

12. Alternate Way to Do Least Squares Alternate Way to Do Least Squares in Multiple Regression in Multiple Regression = + + + Y b b X b X e Model: 0 1 1 2 2 i i i i You want to compute b1. You can do it in two steps and you will get the same number as if you use the MR formula on slide 8. N = = + + X c c X v Step 1: v y 1 0 1 2 1 i i i 1 i i = i 1 N i b 1 2 1 v = + + Y b 1 1 bv e Step 2: i 0 i i i = 1

13. Simplifying Notation with Matrix Algebra Simplifying Notation with Matrix Algebra One equation for each observation = + = + Y b b X + + + + + + + + ... ... Y b b X b X b X b X b X e e 1 0 1 11 2 21 1 1 K K 2 0 1 12 2 22 2 2 K K = + + + + + ... Y b b X b X b X e 0 1 1 2 2 N N N K KN N Combine into matrices 1 1 Y Y X X X X e e b b 1 11 1 1 0 K 2 12 2 2 1 K = + 1 Y X X e b 1 N N KN N K = + Y X b e

14. What We Learned What We Learned How to solve the least-squares problem to fit a MR model. MR estimates differ from simple regression estimates if the right-hand-side variables are correlated with each other. How to apply the MR least-squares formula using a spreadsheet. How to interpret a MR coefficient R2 increases when you add an X variable to a model