统计学Reading 11 相关和回归Reading 12 多元线性回归1. 相关分析:1) 协方差和相关系数的计算;2) 相关系数的检验(ρ = 0): t = r ((n - 2) / (1 – r2))1/2, df = n – 2.2. 线性回归:1) 模型及假设, dependent variable (Y) and independent variable (X).2) 参数估计(min SSE): b i = Cov(X i, Y) / Var(X i), b0 = E(Y) –Σb i * E(X i).3) 回归系数的置信区间和统计检验: t = (b i–βi) / s bi, df = n – k – 1.注：k为independent variable的数目, 不要求计算标准误s bi.3. 线性模型的显著性检验:1) Total sum of squares (SST) = Σ(Y – E(Y))2, Y –观测值;Regression sum of square (RSS) = Σ(y – E(Y))2, y –拟合值;Sum of squared error (SSE) = Σ(Y - y)2;SST = RSS + SSE.3) Determination coefficient R2 = RSS / SST,Adjusted R2 = 1 – (n - 1)/(n – k - 1) * (1 – R2): adjust the impact of additional variables.4) F-检验：F = MSR / MSE, df = (k, n – k - 1).4. 模型前提假设的检验:1) Heteroskedasticity:A) Effect :Unconditional: heteroskedasticity is unrelated to the level of X, no major problem;Conditional: coefficient estimates are not affected, but s.e. and F-test are unreliable.B) Detecting:a) Residual plot: residual = actual – predicted.b) Breusch-pagan test (H0: no heteroskedasticity):n * R resid2 ~ χ2(k), 其中R resid2是residual对X回归的决定系数.C) Correcting: using robust or corrected s.e.2) Serial correlation: residual terms are correlated with each other:A) Effect:Positive serial correlation: underestimating s.e., unreliable F-test;Negative serial correlation: overestimating s.e., unreliable F-test.B) Detecting:a) Residual plot.b) Durbin-Watson test (H0: no serial correlation):DW ≈ 2(1 - r), 其中r是残差的自相关系数, 0 ≤ DW ≤ 4.C) Correcting: corrected s.e. for both serial correlation and heteroskedasticity.3) Multicollinearity: independent variables are highly correlated with each other:A) Effect: individual variables are not significant (large s.e.) but their combination is.B) Detecting:a) None of individual coefficient is significant, but F-test is.b) Correlations between variables (> 0.7).注：两变量间的相关系数并未考虑变量线性组合的相关性，因此低相关系数并不一定意味着不存在multicollinearity.C) Correcting: omit correlated variables or take stepwise regression.4) Model misspecification: 不合适地选取解释变量（实际意义不对或不满足线性模型的前提假设）或不恰当的变量转换等.5. Dummy variable (0 – 1 variable):1) Independent dummy variable: linear regression is appropriate.注：n classes → (n - 1) dummy variable.2) Qualitative dependent variable (Y取0, 1): ordinary regression may not be appropriate, use logit regression model or discriminate model.Reading 13 时间序列分析1. Trend model:1) Linear: x t = b0 + b1t + εt;2) Log-linear: ln(x t) = b0 + b1t + εt, 常用于为增长率建模.2. Autoregressive model (AR):1) AR(p): x t = b0 + b1x t-1 + … + b p x t-p + εp, lag: 1, … , p.2) Covariance stationarity:a) Constant and finite E(X t): mean-reverting level for AR(1) E(X t) = b0 / (1 – b1).Random walks (unit root process): x t = b0 + x t-1 + εt不是平稳过程.b) Constant and finite Var(X t);c) Constant and finite Cov(X t, X t-k):对AR平稳序列, 自相关系数Cor(X t, X t-k) = ρk→ 0.3) Forecasting:a) In-sample (within the range of data) and out-sample forecasts.b) Short time series: more stable (no dramatic change);Long-time series: more reliable.c) Predicting power: root mean squared error (RMSE) on the out-sample data.3. AR模型的检验与修正:1) Nonstationarity:a) Detecting: Dickey-Fuller (DF) test or unit root test:For AR(1): x t– x t-1 = b0 + (b1 - 1)x t-1 + εt, H0: b1 = 1, t-test with modified s.e.b) Correcting - differencing:For random walks: 令y t = x t– x t-1, 有y t = b0 + εt.2) Serial correlation (residual terms should not exhibit serial correlation):a) Detecting:H0: Cor(εt, εt-k) = 0 for any lag k; t = Cor(εt, εt-k) / (1 / T) 1/2, df = T – p - 1.注: T – number of effective observations, T = n – p.一般线性回归中的DW检验不适用于AR模型。

b) Correcting: increase the order p until all serial correlations are removed.3) Seasonality:a) Detecting: seasonal lag k: Cor(εt, εt-(k-1)) = 0, but Cor(εt, εt-k) ≠ 0 (同上述t检验).b) Correcting: include a seasonal lag term: x t = AR(p) + b k x t-k.4) Conditional heteroskedasticity:a) Autoregressive conditional heteroskedasticity model (ARCH):εt2 = a0 + a1 * εt-12 + ηtb) Detecting: H0: a1 = 0.4. Multivariate time series model (Y t ~ X t):1) Premise for regression model:a) Y t and X t are both stationary;b) Y t and X t are both non-stationary but cointegrated.2) Cointegration test:y t = b0 + b1x t + εt, εt is tested for a unit root using DF test.经济学Reading 14 经济增长Reading 20 经济衡量1. Measuring economic activity:1) Measures of economic activity:GNI (国民总收入) = GDP + net property income from abroad;NNI (国民净收入) = GNI – depreciation.2) GDP at current price = GDP at constant price * (1 + GDP deflator).注：GDP平减指数既包含物价变动，也包含GDP组分的变动2. Measuring economic growth:1) Growth rate in real GDP;2) Rule of 70: Approximate years for real GDP to double = 70 / growth rate (percentage).3. Sources of economic growth:1) Preconditions: incentive system – markets, property rights and monetary exchange.2) Key factors driving economic growth:a) Labor: investment in human capital;b) Capital: saving and investment in new capital;c) Entrepreneurial ability: discovery of new technologies.3) Productivity curve: labor productivity (real GDP per labor hour) ~ capital per labor houra) Growth in capital per labor hour – movement along curve, law of diminishing returns;b) Growth in technology – curve shift upwards.c) One-third rule:Labor productivity growth results from capital growth = 1/3 * growth in capital per labor hour;Labor productivity growth results from technology growth = labor productivity growth - labor productivity growth results from capital growth.4. Growth theory:1) Classical growth theory (constant subsistence level): labor productivity rises above the subsistence level → subsistence real wages rise above the minimum required → population explosion → labor productivity is driven back to the subsistence level.2) Neoclassical growth theory (constant target real rate of return = Δreal GDP per labor hour / Δcapital per labor hour): labor productivity rises due to lucky technology changes →increase in real rate of return → more investment and savings, but law of diminishing return holds → real rate of return is driven back to the target rate.注：Population growth is independent of economic growth in the neoclassical model. The technology (and productivity) may stop growing when the real interest rate is equal to the target rate of return.3) New growth theory (continuous economic growth): innovation → higher return →competition → lower return → incentive for innovation.注：Knowledge capital is public goods and it is not subject to the law of diminishing return due to the incentive for innovation.Reading 15 产业政策1. Industry regulation:1) Social regulation: for all industries, such as product quality and safety.2) Economic regulation: pricing policy for the firms within a industry:a) Cost-of-service regulation: set the maximum price;b) Rate-of-return regulation: set the maximum rate of return.2. Effects of regulation:1) Negative effects: higher costs of product response to regulation;Deregulation has negative effects in the short run, but the long-run benefits include lower cost and better products.2) Regulators’ behavior:a) Capture hypothesis: regulators will be captured by the industry that is regulated.b) Share-the-gains, share-the-pains theory: regulators will strive to all three interested parties including legislators, customers and regulated industries.Reading 16 世界贸易1. Comparative advantage:1) Concept: the lowest opportunity cost of a product. For two goods A and B, if one country has comparative advantage in A, the other must have comparative advantage in B.2) Law of comparative advantage: total production and consumption will increase if trade partners export goods with comparative advantage and import goods without advantage.2. Trade restrictions:1) Reasons for trade restrictions:a) Desire by government for tariff revenue.b) Domestic industries to gain protection (infant industry argument, anti-dumping argument).c) Other arguments not supported by economic theory in the long run: protect jobs and high wage levels.2) Trade barriers:a) Tariffs (关税): benefit protected industry and government.b) Quotas: benefit protected industry and import license holders.c) Voluntary export restraints (exporting countries voluntarily limit the quality of goods for exporting): benefit protected industries and export license holders.3. Balance-of-payments (BOP) accounts (国际收支账户) (Reading 17):1) Accounting equation: current account + capital account + official reserve account = 0.2) Current account = export – import: exchange of goods and services.3) Capital account = investment in – investment out, including equity and debt;Borrowing (capital account > 0) to finance a current account deficit.4) Official reserve account:If current account + capital account > 0 (收支盈余), official reserve account < 0 (而外汇储备增加).Reading 18 汇率1. Exchange rate:1) Quotation, currency appreciation and depreciation:a) Direct quotes: DC/FC, or FC:DC (注意后者不表示相比);b) Indirect quotes: FC/DC, or DC:FC.注：一般汇率S均指直接标价，S↑外币升值（本币贬值）。