Relationship vs sampleestimated relationship write

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-pop. Relationship vs. sample(estimated) relationship -write the sample(estimated) relationship as an equ. -Yi=b0+b1xi+ei, where i is observation # and b- and b1 are OLS (ordinary least square) estimates -The fitted line is y(hat)=b0+b1X -How can we find the values for b0 and b1? How can we estimate them? -Find b0 and b1 such that sum of squared residuals is minimized GRAPH Multiple regression -Household fuel consumption is affected by fuel price and residential location (2 variables) -Let Y denote fuel cons., and X1 denote fuel price and X2 denote residential location, the assumed relationship is Y=β0+β1X1+β2X2+ε -In this way, you incl. All relevant variables that may influence Y Interpretation of β1 and β2 -β1 is the expected change in Y from a unit change in X1, holding X2 fixed: β1=ΔE[Y|X1,X2]/ΔX1 | ΔX2=0 -β2 is the expected change in Y from a unit change in X2, holding X1 fixed: β2=ΔE[Y|X1,X2]/ΔX2 | ΔX1=0 -Travel Demand -Individuals demand travel for commute to work, shop, leisure, etc -To measure demand, we use: -Quantitative measures: (distance traveled, fuel usage) -Qualitative choices: (decision to travel, travel modes) -Other relevant choices: (vehicle-type, trip timing) -Price (cost) influences travel demand Demand function when consumer choice is continuous (quantitative measure) -How long (in miles) would the consumer travel by car in a year? -Let T denote Vehicle Mileage Traveled (demand for travel) -Let P denote fuel price -The demand function is written T=f(p) -Does T decline w/P, so that f’(p)<0? By how much? GRAPH Relationship b/t price and VMT (Vehicle Mileage Traveled) -Household i’s travel demand T(VMT) is determined by -Ti= β1+β1Pi+β2yi+εi -i is household id with i=1,2,...n, -yi is household i’s income -εi is household-specific factors influencing T -Conditional expectation: E[T|p,y]=β0+β1p+β2y
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-expected VMT for household with p and y Interpretation of coefficients -β1 gives exp. Change in T from unit increase in p, holding y fixed β1=ΔE[T|p,y]/Δp | Δy=0 -β2 gives exp change in T from a unit incr. In y, holding p fixed β2=ΔE[T|p,y]/Δy | Δp=0 A numerical ex. of estimation results -Ex. If you estimated following equation: T=22.5-485p+0.12y - How will you interpret the coefficients? - -485 means that a household is expected to decrease (demand for it) VMT by 485 mi when fuel price incr by 1 dollar - 0.12 means that a household w/1 dollar higher income is expected to drive 0.12 vehicle mi more than another household Motivation for a different model -T is a continuous(quantitative) variable in the demand function -Consumer’s VMT is continuous -Consumer chooses how much/many to consume -Many transportation-related choices are discrete(categorical) -They often choose among several alternatives -Estimates of travel demand involves estimating demand for discrete(categorical) choices Terminology
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