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EEL6323-S10-HLec01-intro-2spp

Course: ESI 6323, Spring 2009
School: University of Florida
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6323 Advanced EEL VLSI Design Spring 2010 Section: 6963 Lecture 0: Introduction Course Outline CMOS Trends and Scaling 1 Practical Information Instructor Rizwan Bashirullah Office: 527 NEB, E-mail: rizwan@ufl.edu Tel: (352) 392-0622, Fax: (352) 392-8381 Admin Janet Holman, 526 NEB holman@ece.ufl.edu Practical Information Class Meeting time: MWF 10:40-11:30 Location: NEB 100 Website:...

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6323 Advanced EEL VLSI Design Spring 2010 Section: 6963 Lecture 0: Introduction Course Outline CMOS Trends and Scaling 1 Practical Information Instructor Rizwan Bashirullah Office: 527 NEB, E-mail: rizwan@ufl.edu Tel: (352) 392-0622, Fax: (352) 392-8381 Admin Janet Holman, 526 NEB holman@ece.ufl.edu Practical Information Class Meeting time: MWF 10:40-11:30 Location: NEB 100 Website: http://www.icr.ece.ufl.edu/teaching/EEL6323S10/6323home.htm (password protected) Instructor Office Hours: MW 11:30 - 12:30pm TA Qiuzhong Wu (qiuzhongwu@ufl.edu) Chung-Ching Peng (cpeng@cise.ufl.edu) Office hours: TBA 2 Class Material Required Textbooks Neil H.E. Weste, David Harris, CMOS VLSI Design, A Circuits and Systems Perspective, 3rd Edition, Pearson, Addison-Wesley, 2005. ISBN 0-321-14901-7 Jan. M. Rabaey, A. Chandrakasan, and B. Nikolic, "Digital Integrated Circuits, A Design Perspective," 2nd Edition, Prentice Hall, 2003. ISBN 0-13-090996-3 In addition, handouts developed by instructor may be downloaded from class www site Class Material Journal/Conference References Journal of Solid State Circuits (JSSC), TVLSI, CAS-I and II ISSCC, VLSI Symposium, CICC, ISCAS Web Links IEEE Explorer 3 Course Goals To develop a basic understanding of CMOS integrated circuit design To develop proficiency in analysis, design and implementation of CMOS circuits To develop a basic understanding of design considerations to maximize chip success Course Topics CMOS Overview Scaling Trends and Process Technology Gates Transistor Models DC/Transient and Delay Logical Effort Circuit Simulation Static, Dynamic Variability and Circuit Pitfalls Low power design Power dissipation Low power techniques and trends Datapaths Adders Multipliers Timing Digital Flow Design methodology Synthesis, Place and Route, verification Circuit Families Clock Distribution and Generation Skew Tolerant Design Latches and Flip-flops PLL/DLL Signaling and power Interconnects, Packaging and I/O Power Delivery and Decoupling Memory 4 Digital IC Flow RTL to layout Behavioral Language Synthesis Place and route Verification -- right-shift arithmetic for 8-bit signed integer rsa: process (a, shamt) variable fill : std_logic_vector(1 downto 0); variable temp : std_logic_vector(4 downto 0); begin for i in 0 to 3 loop fill(i):=1 and a(3); end loop; if shamt(0)=1 then temp := fill(0) & a(7 downto 1); end if; if shamt(1)=1 then temp := fill(1 downto 0) & temp(7 downto 2); end if; if shamt(2)=1 then out <= fill(3 downto 0) & temp(7 downto 4); end if; end process; Grading Policy Problem Sets (5): 30% Midterms (2): 20% Each Final Project (1): 30% Grading policy is subject to change 5 Projects USB controller and IO interface EPC Gen 2 RFID protocol or Smart Card Ultra low power embedded processor and communication interface Academic Honesty All students admitted to the University of Florida have signed a statement of academic honesty committing themselves to be honest in all academic work and understanding that failure to comply with this commitment will result in disciplinary action. This statement is a reminder to uphold your obligation as a student at the University of Florida and to be honest in all work submitted and exams taken in this class and all others. Students requesting classroom accommodation must first register with the Dean of Students Office. The Dean of Students Office will provide documentation to the student who must then provide documentation to the instructor when requesting accommodation. 6 Lecture 1: Introduction Moores Law CMOS Trends 1975 1974 1973 1972 1971 1970 1969 1968 1967 1966 1965 1964 1963 1962 1961 Electronics, April 19, 1965. 1960 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1959 LOG2 OF THE NUMBER OF COMPONENTS PER INTEGRATED FUNCTION Moores Law In 1965, Gordon Moore noted that the number of transistors on a chip doubled every 18 to 24 months. He made a prediction that semiconductor technology will double its effectiveness every 18 months Digital Integrated Circuits, 2nd Ed. 2003 Prentice Hall/Pearson 7 Moores law in Microprocessors Transistors (MT) 1000 2X growth in 1.96 years! 100 10 486 1 P6 Pentium proc 386 286 0.1 8086 8085 Transistors on Lead Microprocessors double every 2 years Transistors on Lead Microprocessors double every 2 years 0.01 8080 0.001 8008 4004 1970 1980 1990 Year 2000 2010 Courtesy, Intel Digital Integrated Circuits, 2nd Ed. 2003 Prentice Hall/Pearson Memory Trends 32 word lines x 1024 blocks 16896 bit lines 10.7mm Charge pump 2kB buffer & cache 1Gbit flash memory 11.7mm From [Nakamura02] 8 Feature Size Human hair, 100m Amoeba, 15m Red blood cell, 7m AIDS 0.1m [Source: virus, Gordon Moore, Intel, ISSCC] Die Size Trends 9 Die Size Growth Die size (mm) 100 10 8080 8008 4004 8086 8085 1 1970 286 386 P6 486 Pentium proc ~7% growth per year ~2X growth in 10 years 1980 1990 Year 2000 2010 Die size grows by 14% to satisfy Moores Law Die size grows by 14% to satisfy Moores Law Courtesy, Intel Digital Integrated Circuits, 2nd Ed. 2003 Prentice Hall/Pearson Frequency Frequency (Mhz) 10000 Doubles every 2 years 1000 100 486 10 8085 1 0.1 1970 8086 286 P6 Pentium proc 386 8080 8008 4004 1980 1990 Year 2000 2010 Lead Microprocessors frequency doubles every 2 years Lead Microprocessors frequency doubles every 2 years Courtesy, Intel Digital Integrated Circuits, 2nd Ed. 2003 Prentice Hall/Pearson 10 Processor Supply Voltage [Source: Gordon Moore, Intel, ISSCC] Static Power VDD decreases Save dynamic power Protect thin gate oxides and short channels No point in high value because of velocity sat. Vt must decrease to maintain device performance But this causes exponential increase in OFF leakage Major challenge [Source: Gordon Moore, Intel, ISSCC] Processor Power (Watts) Active and Leakage 11 Power Dissipation Power (Watts) 100 P6 Pentium proc 10 8086 286 1 8008 4004 486 386 8085 8080 0.1 1971 1974 1978 1985 1992 2000 Year Lead Microprocessors power continues to increase Lead Microprocessors power continues to increase Courtesy, Intel Digital Integrated Circuits, 2nd Ed. 2003 Prentice Hall/Pearson Power density Power Density (W/cm2) 10000 1000 100 Rocket Nozzle Nuclear Reactor 8086 10 4004 Hot Plate P6 Pentium proc 8008 8085 386 286 486 8080 1 1970 1980 1990 2000 2010 Year Power density too high to keep junctions at low temp Power density too high to keep junctions at low temp Courtesy, Intel Digital Integrated Circuits, 2nd Ed. 2003 Prentice Hall/Pearson 12 Multi-core: 80 Tile Processor Intel Device Enhancements TiSi2 n+ S Al Gate D Gate NiSi Gate Si1-xGex Si Substrate (a) 1970s LDD (b)1980s High-k Highgate dielectric Gate Silicide added Channel Strained silicon Strained channel (c) 2003 Transistor New transistor structure Source: Intel 13 Gate Stack Traditional Oxide Scaling XG=LG/45 70nm node, XG~2nm (6-7 atomic layers) Increased gate leakage Introduce High-k Gate dielectrics Thicker oxide to reduce leakage Use high dielectric constants to maintain same capacitance per unit area hafnium oxides HfO2 kx~30-40 zirconium oxides ZrO2 kx~25 Equivalent Oxide Thickness (EOT) EOT=(kSiO2/kx)tx kSiO2 ~3.9 tx = physical oxide thickness Strained Silicon and Silicides Higher mobility Increases drive current and transistor speed Implant SiGe to stretch the Si lattice Yields up to 70% increase in mobility or 30% improvement in performance Silicide Polysilicon gate and Source/Drain regions have high resistance Deposit metal on polygate and/or S/D regions to decrease resistance (Tantalum, molybdenum, titanium, cobalt) Polycide: only the gate is silicided Salicide (self-aligned silicidization): both polygate and S/D regions are silicided 14 LDD and Isolation LDD Lightly doped drain Reduces the electric field at the drain junction Improves device immunity to hot electron damage LDD is shallow and increases series resistance (decreases performance) Silicon Nitride spacer along the edge of gate serves to mask n+/p+ S/D regions LOCal Oxidation of Silicon (LOCOS) to isolate devices LOCOS produces Birds beak caused by lateral diffusion of SiO2 Limits packing density due to extended transition between thin and thick oxides Shallow Trench Isolation (STI) 180nm and below Improved packing density 90nm Process Interconnects M7 Metal lines Al M6 Cu M5 M4 Insulating dielectric M3 M2 M1 (low-k) (low- Interconnects Source: Intel 15 Low-K Dielectrics and Copper Interconnects Low-K is used for Inter Layer Dielectrics (ILD) to decrease interconnect capacitance SiO2 dielectric constant k=3.9-4.2 Addition of Fluorine to SiO2 forms fluorosilicate glass (FSG) and decreases k to ~3.6 Replace Al interconnects with Cu up to 40% decrease in resistance Process enhancements Isolation, Reliability, Gate stack, Strained Si Low-k and copper interconnects Other enhancements Multiple Vt transistors Low-Vt and high-Vt Multiple Oxide thickness Core logic vs IO logic Thick metal options Triple Well process 16 Summary Number of transistors doubles ~18-24mo Die size increase by 14% every 2 years Frequency doubles every 2 years But power limit Leakage power is increasing as TOX decreases (20% or more)new devices Supply voltages have been steadily decreasing But will remain at 1V or so Minimizing Power is essential Different architectures, slower speeds, power gating 17
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University of Florida - ESI - 6323
Homework 1 Find the following for 180nm, 130nm, 90nm, 65nmand 45nm CMOS technologiesDue Wed Jan 20, 2009 Effective channel length Equivalent and physical oxide thickness Supply voltage (Vdd) Draw the layout for the following Latch (use TSMC0.25um
University of Florida - ESI - 6323
Lecture 3: LayoutCMOS EnhancementsScalable rulesPoly orderingDesign PartitioningFloorplanningLayout Min. feature size expressed in terms of = f/2 E.g. = 0.3 m in 0.6 m process Lambda rules are conservative All dimensions rounded to integer multi
University of Florida - ESI - 6323
Lecture 4: DC and Transient AnalysisDC AnalysisSkewed GatesLogic Levels and Noise MarginsTransient ResponseDelay Estimation Reading: Ch. 2Load Line AnalysisFor a given Vin:Region Plot Idsn, Idsp vs. Vout Vout must be where |currents|are equal
University of Florida - ESI - 6323
EEL 6323 Advanced VLSI Design - Spring 2011Instructor: R. BashirullahTA: Qiuzhong Wu (qiuzhongwu@ufl.edu)(Due Friday April 27, 2010)The goal of the project is to study one of the topics specified and design an architecture which consumeslow power, is
University of Florida - ESI - 6323
Lecture 5: Logical Effort Logical Effort (g, h, p) Examples Reading: Ch. 4IntroductionChip designers face a bewildering array of choices What is the best circuit topology for a function? How many stages of logic give least delay? How wide should t
University of Florida - ESI - 6323
Lecture 6: Multistage Logic Networks Multistage Logic Networks Reading: Ch. 4Multistage Logic Networks Path Logical Effort10xg1 =g2 =yg3 =zg4 =h1 =h2 =h3 =h4 =201Multistage Logic Networks Path Electrical Effort10xg1 = 1g2 = 5/3yg
University of Florida - ESI - 6323
Lecture 8: Circuit Simulations Circuit Characterization (Brief) Reading: Ch. 5IV curves: Basic Shapes IDS vs VDS (two regions) Linear (Low VDS) Effective Resistance Saturated (High VDS) Current, gm, gds IDS vs VGS Linear IDS (above threshold) L
University of Florida - ESI - 6323
Lecture 9: Power Dissipation Power and Energy Dynamic and Static Power Leakage Reading: Ch. 4Power Trends Trend in CMOS power dissipation Proportional to chip area and frequency[M. Horowitz, EE246, Stanford Univ]1Power Issues Three reasons why
University of Florida - ESI - 6323
Lecture 11: Power Dissipation Low Power DesignThroughput oriented designClock gatingLeakage reduction techniquesMulti-processing trendsTypes of Processing Fixed-rate Processing (i.e. Signal processing formultimedia or communications) Stream based
University of Florida - ESI - 6323
Phases of Design Flow Phase 1: Design Planning From application requirements to specificationsOverview of Digital IC Design Flow Phase 2: Design Implementation and Verification From SPEC to layout (GDSII) Phase 3: Design Review and Tape-outQiuzhong
University of Florida - ESI - 6323
Synthesizable RTL CodingRTL Coding and Simulation When writing RTL, Always think of Hardware! Think of Synchronous Hardware Synchronousdesign can run smoothly during synthesis, test,simulation, and layout. Think of RTL, writing in RTL coding style,
University of Florida - ESI - 6323
Logic Synthesis Logic Synthesis= Translation+ Optimization+ MappingLogic Synthesis2Gate-Level OptimizationLogic Synthesis Flow34Design Compiler ProcedureLogic Synthesis Input/Output56Describing Design EnvironmentDesign Environment Beware th
University of Florida - ESI - 6323
Lecture 15: Circuit Families Pseudo-nMOS Logic Dynamic, Domino, NP-Domino CVSL, LEAN, CPLDelay What makes a circuit fast?I = C dV/dt -&gt; tpd (C/I) Vlow capacitancehigh currentsmall swing Logical effort is proportional to C/I pMOS are the enemy!
University of Florida - ESI - 6323
Lecture 16: AddersSingle-bit AdditionCarry-Ripple AdderCarry-Skip AdderCarry-Select AdderCarry-Lookahead AdderTree Adder Reading: Chapter 10, W&amp;HSingle-Bit AdditionHalf AdderS =ABCout = ABAB0ABFull AdderAS=Cout =CoutCoutCSSCout S
University of Florida - ESI - 6323
Lecture 17: AddersSingle-bit AdditionCarry-Ripple AdderCarry-Skip AdderCarry-Select AdderCarry-Lookahead AdderTree Adder Reading: Chapter 10, W&amp;HCarry Generation and PropagationDefine 3 new variables that depend only on A and B Generate: Cout =
University of Florida - ESI - 6323
Lecture 18: DatapathsMultipliersShiftersComparatorsCountersLFSRsMultiplication Example:1100 : 12100101 : 510110000001100000000111100 : 6010multiplicandmultiplierpartialproductsproduct M x N-bit multiplication Produce N M-bit partial p
University of Florida - ESI - 6323
Lecture 19: Clock DistributionClock distribution trendsDistribution networksClock PowerClock SkewTiming Definitions Source: Ch 7 J. Rabaey notes, Weste and Harris Notes, S. Russu, ISSCC,Clocking Synchronous systems use a clock to keep operations
University of Florida - ESI - 6323
Lecture 20: Sequential Circuits Sequencing Elements Simple Latch/FF Timing Definitions Source: Ch 7 (W&amp;H)SequencingUse flip-flops to delay fast tokens so they move throughexactly one stage each cycle.Inevitably adds some delay to the slow tokensC
University of Florida - ESI - 6323
Lecture 21: Sequential CircuitsSetup and Hold timeMS FF Power PCPulsed FF HLFF, SDFF, SAFFSource: Ch 7 J. Rabaey notes, Weste and Harris NotesReview: Timing Definitions TCQ: Propagation Delay from Ck to Q, assuming D has been setearly enough relat
University of the East - BUS - 101
Active CellIn a worksheet, the cell with the black outline. Data is always entered into the active cell.Column LetterColumns run vertically on a worksheet and each one is identified by a letter in the columnheader.Formula BarLocated above the worksh
Grand Canyon - MKT - 607
Week 3 DB Due by day 3With respect to the articles assigned and any otherarticles/research that you are able to draw upon, do marketershave an obligation to avoid marketing to vulnerable consumers asdefined by Smith and Cooper-Martin in &quot;Ethics and Ta
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Week 3 DB due by day 5Explain the reason for positioning and repositioning products. Choose a product with which youare familiar, preferably one in your industry, and explain how it might berepositioned. Indicateits current position in the market, a de
Grand Canyon - MKT - 607
Week 4 DB Due by day 3As stated in this week's reading, &quot;Whether a company grows,survives, and makes a profit could depend upon how theirproducts or services are defined.&quot; What does this statement meanin to a company or to consumer perception?Greetin
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Week 5 DB Due by day 3Define integrated marketing communications (IMC) and discuss theimportance of teamwork in achieving a successful IMC effort.Provide examples of how this concept would apply to a specificorganization. Use the Schultz and Kitchen a
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Week 5 DB Due by day 5Identify possible ethical issues involved with the dominance oflarge retailers (e.g., Wal-Mart, Home Depot, etc.). Explain yourposition regarding these issues and propose possible solutionsfor avoiding such issues if possible.Gr
Rutgers - PHARM - 300
! http:/www.pitt.edu/~super1 JIT http:/www.pitt.edu/~super1 : . . . . :
Rutgers - PHARM - 300
Medical Aspects of Blast InjuriesAssistant Professor of Emergency Medicine Mayo Clinic sztajnkrycer.matthew@mayo.eduMatthew D. Sztajnkrycer, MD, PhDAmado Alejandro Bez MD Mscbaez.amado@mayo.eduLearning Objectivess Discuss the epidemiology of blast
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Disaster and Multi-Casualty TriageAmado Alejandro Bez MD MSc Matthew Sztajnkrycer MD PhDLearning Objectives Describe the key elements of disaster triage Understand the basic principles of Mass Casualty Triage (START)Performance Objectives At the end
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Kansas 9/14/04 TornadoSpring storm and tornadoes in KansasSatellite image taken Thursday at 11:15 p.m. EDT.
Rutgers - PHARM - 300
TornadoesScott R. Lillibridge, M.D.Centers for Disease Control &amp; PreventionINTRODUCTIONBackground and Nature of the ProblemTornadoes are funnel-shaped wind storms that occur when masses of air with differing physical qualities (e.g., density, tempera
Rutgers - PHARM - 300
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Public Health Consequences of Earthquakes. Part II.Eric K. Noji, M.D., M.P.H.Centers for Disease Control and Prevention Washington, D.C.PREVENTION AND CONTROL MEASURESUntil earthquake prevention and control measures are adopted and mitigation actions
Rutgers - PHARM - 300
Access to and Need for Counseling Among Children after the September 11th Attacks on the World Trade CenterGerry Fairbrother, PhDNew York Academy of MedicinePresentation to the 2003 Pediatric Academic Societies Annual Meeting May 3-6, 2003 Seattle, WA
Rutgers - PHARM - 300
Laboratory Criteria for Identification of B. anthracisssFrom clinical samples, such as blood, cerebrospinal fluid (CSF), skin lesion (eschar), or oropharyngeal ulcer Encapsulated gram-positive rods on Gram stain From growth on sheep blood agar: Large g
Rutgers - PHARM - 300
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LSE - ECON - 201
EC201Lent TermErik EysterFinding a General Competitive Equilibrium in an ExchangeEconomy.1. In an exchange economy, Agent A has endowment (2, 2) and preferences uA (xA , xA ) = xA1215xA22, while Agent B has endowment (2, 1)and preferences uB
LSE - ECON - 201
EC201Lent 2011Erik EysterSecond Worked Example of General Competitive EquilibriumIn this worked example, we use the example of two wheat farmers tradingwheat now for wheat later to explore interest rates set by a competitivemarket.Two farmers trade
LSE - ECON - 201
MACROECONOMICS TEST 1 (Dr Ashwin MOHEEPUT)GENERAL KNOWLEDGE QUESTIONSQuestion 1Suppose a car manufacturer is choosing between two production options. It can produce 100cars with 200 workers and 50 machines, or it can produce 166 cars with 300 workers
LSE - ECON - 201
EC201Lent TermErik EysterFinding a General Competitive Equilibrium in an ExchangeEconomy.1. In an exchange economy, Agent A has endowment (2, 2) and preferences uA (xA , xA ) = xA1215xA22, while Agent B has endowment (2, 1)and preferences uB
LSE - ECON - 201
EC201Lent 2011Erik EysterSecond Worked Example of General Competitive EquilibriumIn this worked example, we use the example of two wheat farmers tradingwheat now for wheat later to explore interest rates set by a competitivemarket.Two farmers trade
LSE - ECON - 201
Dear EconomistsAs the exams are approaching and we all start digging into past exam papers and old supervisions,Economics Society invites everybody to join our Unofficial Solutions Database.We are going to start accumulating student-made answers on our
LSE - ECON - 201
Dear EconomistsAs the exams are approaching and we all start digging into past exam papers and old supervisions,Economics Society invites everybody to join our Unofficial Solutions Database.We are going to start accumulating student-made answers on our
LSE - ECON - 201
MACROECONOMICS TEST 1 (Dr Ashwin MOHEEPUT)GENERAL KNOWLEDGE QUESTIONSQuestion 1Suppose a car manufacturer is choosing between two production options. It can produce 100cars with 200 workers and 50 machines, or it can produce 166 cars with 300 workers
LSE - ECON - 201
DEMAND FOR INSURANCEA consumer begins with initial wealth w and faces probability p of a loss of L in anaccident. She has concave utility of wealth function u(w) and strictly decreasing marginalutility. Lets call her risky no-insurance endowment X .Th
LSE - ECON - 201
models.CSystematic 2 T O O L S Fis R astlyM P A R A T I and S T A T I C S much less likely to occur. Indeed, suppose34H A P T E R 1 elimination O v C O simpler, V E errors arean economic model reduces to a system of two linear equations in two unknowns
LSE - ECON - 201
Responses to Serial CorrelationIf the disturbances are serially correlated OLS parameter estimates are unbiased OLS parameter estimates are inefficient the OLS standard errors are incorrect (there aremethods which correct for this)There are two poss
LSE - ECON - 201
Regression using time series data stationary I(0) seriesProcedures to be adopted depend on whether timeseries are stationary or nonstationaryHence we need to pretest data using DF and ADFtests Modelling I(0) series - ie all the series are stationary
LSE - ECON - 201
Testing time series for unit rootsWe know that a random walk is a particular type ofAR(1) processwith = 1xt = xt-1 + etHence to test whether xt is a random walk (with zeromean), we could estimatext = xt-1 + etand testH0 : = 1nonstationarityagai
LSE - ECON - 201
Nonstationary stochastic processesStationary processes satisfyE[xt] = Var[xt] = 2 &lt; Cov[xs,xt] = t-sall independent of tMany economic series do not satisfy these conditionsE[GDP1970] &gt; E[GDP1870]Hence they are nonstationary and cannot berepresent
LSE - ECON - 201
Time series - processes andrealisationsWith random sampling the key concepts are populationand sampleThe population can be real (Census) or hypothetical(continuous distribution)Sample statistics are (imprecise) estimates oftheunderlying population
LSE - ECON - 201
Introduction to time series analysisIn many cases (especially in macroeconomics) a sampleconsists of a set of observations measured over timeSuch data cannot be treated as a random sample - infact we need new concept of population and sampleIn time s
LSE - ECON - 201
Variables usedAGEWAGELNWAGEIndividuals ageHourly wage (in $)Log of wageOCC1Categorical variable for occupational category (seebelow)Categorical variable for industrial category (see below)1 if union member, 0 otherwiseIND1UNIONGRADEMARRIED
LSE - ECON - 201
Introduction to Diagnostic TestingIn a regression equation the N observations generate K regression coefficients N residuals with (N - K) distinct components(because ei = 0 etc)These residuals should not have a systematic patternDiagnostic statistic
LSE - ECON - 201
Forecasting and regressionFor a simple regression, given OLS estimates of the regression coefficients e, e the forecast value of X, XpThe point forecast isyp = e + eXp since E[up] = 0The forecast error isep = y - yp= ( + Xp + up) - (e + eXp)= ( -
LSE - ECON - 201
Model SpecificationWe do not know from economic theory which regressorsshould be included in a multiple regression. Thischoice is a form of hypothesis testingWe distinguish between nested and non-nested modelsModel 1 Y = + X + Z + U1Model 2 Y = + X
LSE - ECON - 201
Multiple Regression: Goodness of FitWe can decompose Var(Y) using a multiple regressionyi = + xi + zi + uiVar(Y) = cfw_(e)2Var(X) + 2eeCov(X,Z) + (e)2Var(Z)+ Var(Ue)(Other terms such as Cov(e, X) and Cov(X,Ue are 0)As in simple regression, this deco