Hendrik Jürges
email: juerges@mea.uni-mannheim.de.

Self-assessed health, reference levels, and mortality (Self-assessed health and mortality)

Keywords: Health Production: Nutrition, Mortality, Morbidity, Substance Abuse and Addiction, Disability, and Economic Behavior Self-assessed health, Response bias, Mortality Self-assessed health, Response bias, Mortality. JEL-Codes: C42, I12

The paper studies the relationship between self-assessed health and subsequent mortality in the German Socio-Economic Panel. Specifically, I examine whether socio-economic characteristics of respondents have an effect on mortality, conditional on self-assessed health. Such conditional effects are shown to exist for various covariates, including age, income, and wealth. These findings question the comparability of self-assessed health across different socio-economic groups.

Introduction

Self-assessed health (SAH) is probably the most common measure of health in general purpose surveys and often the only available indicator of the respondents' health.

Respondents are asked to give an overall judgement of their health: "How is your health in general?". Answers are usually recorded on 5-point scales ranging from "excellent" to "poor" or "very good" to "very bad". The popularity of this measure for applied economists stems from the fact that it lends itself to the treatment of health as a latent, uni-dimensional variable (see [START_REF] Wilson | Work and the accommodation of chronic illness: A re-examination of the health-labour relationship[END_REF], for a critical review). It has been widely used as an outcome variable in studies of social influences on health (e.g. [START_REF] Buckley | The transition from good to poor health: an econometric study of the older population[END_REF]2004[START_REF] Case | Broken down by work and sex: how our health declines[END_REF][START_REF] Soydemir | The impact of religiosity on selfassessments of health and happiness: evidence from the US Southwest[END_REF], or as an explanatory variable in models of labour supply and retirement (e.g. [START_REF] Disney | Ill-health and retirement in Britain: A panel data-based analysis[END_REF][START_REF] Larsen | The Impact of Health on Individual Retirement Plans: a Panel Analysis comparing Self-reported versus Diagnostic Measures[END_REF][START_REF] Wang | Semi-parametric estimation of the effect of health on labour force participation of married women[END_REF].

Much of the debate about using self-assessed health in applied economic centres around its subjective, and thus potentially biased, nature [START_REF] Bound | Self-Reported versus Objective Measures of Health in Retirement Models[END_REF][START_REF] Dwyer | Health problems as determinants of retirement: are self-rated measures endogenous?[END_REF], and the use of more "objective" measures, such as diagnosed conditions or subsequent mortality, is called for. The purpose of the present paper is to study the nature of the relationship between self-assessed health (SAH) as a subjective health indicator and mortality (as an objective health indicator) in the German Socio-economic Panel (SOEP). The relationship between self-assessed (and also, for that matter, diagnosed) health and future mortality is of course not perfect. Since self-assessed health is a measure of an individual's overall health status, not every chronic condition that affects self-assessed health is life-threatening or life-shortening. For instance, a well-known finding in epidemiology is that women are more likely to suffer from chronic diseases but are less likely to die at each age (see [START_REF] Case | Sex differences in morbidity and mortality[END_REF], on sex F o r P e e r R e v i e w 2 differences in health). Further, not every future mortality risk is known to the respondent, and even if it was known, it would not necessarily affect self-assessed health. Despite these conceptual shortcomings, previous research has shown that there is a strong independent relationship between self-assessed health and subsequent mortality (see Idler & Benyiamini, 1997;Benyiamini & Idler, 1999).

This paper is one of the first to use the SOEP for the analysis of self-reported health and subsequent mortality. To the best of my knowledge, there is only one other study based on the SOEP that deals with this issue [START_REF] Schwarze | Self-rated health and changes in selfrated health as predictors of mortality. First evidence from German panel data[END_REF].

However, Schwarze et al. do not address the comparability of self-rated health across different socio-economic groups. But comparisons of self-reports of health across different populations (e.g., across countries, languages, sexes, education and income levels, etc.) are only possible if these measures are in fact comparable. Mortality is a fairly objective indicator of health, and the relationship between self-reported health status and subsequent mortality contains information about differences in self-reports that are not necessarily differences in true health states. One major concern with selfassessed health is that respondents may have different response styles or different reference points against which they judge their health. This gives rise to a fundamental identification problem, namely to distinguish differences in true health from differences in reporting behaviour.

For instance, a common finding is that older respondents tend to have a "milder" view of their health, i.e. they tend to rate their health as better than otherwise comparable younger respondents [START_REF] Groot | Adaption and scale of reference bias in self-assessments of quality of life[END_REF][START_REF] Van Doorslaer | Does inequality in self-assessed health predict inequality in survival by income? Evidence from Swedish Data[END_REF]. Most probably, this happens because health declines in general with age, so that the perception of what is "good health" also changes when individuals get older (the idea of adaptation to [START_REF] Gjerde | On adaptation and the demand for health[END_REF]). One consequence of adaptation is that the decrease of self-reported health in age measured in surveys tends to understate the decline in true health. Another example is socio-economic status. The measurement of social inequalities in health can be severely biased if the response behaviour of rich and poor individuals differs. For example, many studies have shown that unemployment has a lasting effect on general life satisfaction [START_REF] Winkelmann | Why are the unemployed so unhappy?[END_REF][START_REF] Clark | Scarring: the psychological impact of past unemployment[END_REF]. This might influence response behaviour in surveys, leading respondents to a more pessimistic view of their own health than they might otherwise have. If this is the case, effects of unemployment on health will be overstated.

The fundamental assumption behind this view is that there is such thing as a "true" and comparable health status. Conceptually, true health is considered as a continuous, latent (i.e., unobservable) variable. When respondents answer survey questions about their health, they assess their true health (possibly with measurement error; see [START_REF] Crossley | The reliability of self-assessed health status[END_REF] and project this value onto the scale provided by the survey researcher.

Equivalent econometric formulations are the ordered logit or probit models. It is the task of the researcher to "rescale" the respondents answers if response styles or reference points differ across individuals. This implies that one must be willing not to accept people's own judgements as absolute [START_REF] Sen | Health: perception versus observation[END_REF]. The question that follows immediately is on which basis to rescale individual judgements.

A very recent attempt to design a common comparable scales (not only for health) is the anchoring vignette approach [START_REF] King | Enhancing the Validity and Cross-Cultural Comparability of Measurement in Survey Research[END_REF]. Vignettes are short descriptions of persons in different health states, which respondents are asked to judge on the same scale as they are asked to judge their own health. Respondents are 4 explicitly asked to think about the vignette persons as people of their own age and background. The idea is thus that respondents put themselves in the shoes of the vignette persons. An example vignette for mobility is: "John is able to move his arms and legs, but requires assistance in standing up from a chair or walking around the house. Any bending or lifting is very difficult. How would you rate his mobility? Very good, good, moderate, bad, or very bad?" In a typical vignette study, respondents would be asked to answer 3 to 7 vignettes that describe different degrees of mobility, from marathon runners to quadriplegics. If responses are consistent across vignettes and self-ratings, i.e. if the same scale is used by the respondent, it is possible to recover the respondents' individual reference points. The main problem of vignettes is that none have yet been developed for general health, i.e. the self-report measure that I examine in the present paper.

Another approach to rescale the respondent's answers is to use other, presumably more objective measures of health, e.g., the self-reported presence of chronic conditions, functional limitations, problems with activities of daily living, or (disability) weighted indices combining these dimensions (e.g. [START_REF] Groot | Adaption and scale of reference bias in self-assessments of quality of life[END_REF][START_REF] Lindeboom | Cut-point shift and index shift in selfreported health[END_REF]). To examine whether there is reporting bias, subjective health is regressed on objective health and the relevant covariates, possibly in interaction with the objective health measure. The coefficient of a covariate would tell us if certain types of individuals (e.g., men) tend to overstate or understate their health relative to other types of individuals.(e.g., women). If that is the case, one can carry the analysis further and calculate counterfactual health distributions. For instance, one can calculate how the health of men and women compares if we assume that men behave like women (at least when they answer surveys). However, the possibility that self-reports of chronic conditions are also subject to measurement error is often overlooked in studies of this kind [START_REF] Baker | What do self-reported, objective, measures of health measure?[END_REF].

The data used in the present study has only few contemporaneous quasi-objective measures of health, so that it is not possible to follow that approach. Technically, one could use future mortality as an objective health measure as a regressor. However, there are some obvious drawbacks: Right-censoring and sample attrition make mortality an unattractive right-hand side variable. About 20 percent of the sample is lost due to attrition between 1992 and 2003 and of the remaining cases, about 75 percent are rightcensored. Both problems can be much more easily dealt with if mortality is put on the left hand side of the equation. I will thus estimate duration models, regressing mortality on self-reported health and a number of other relevant variables, such as age, sex, or income. Additional specifications will also include interaction effects between selfassessed health and the other covariates.

If self-assessed health is an unbiased measure of health, other covariates should have only negligible independent effects on mortality. If a strong and significant effect of a covariate is found, this might indicate response bias with respect to that variable. For instance, a positive coefficient of some variable X on mortality, conditional on selfassessed health, indicates that respondents with high value of X die earlier than those with low values of X. Put differently, individuals with high X tend to over-value their health compared to individuals with low X. Thus the coefficient of X provides a measure of the response bias associated with X. The basic assumption behind this interpretation is that there are no unobserved factors that are correlated with X. This assumption is violated if, conditional on current self-reported health, future events are systematically linked to the explanatory variables. For example, men might have a F o r P e e r R e v i e w 6 higher probability to die from non-natural causes or to get a disease that kills fast (e.g. lung cancer or a heart attack). If the cause of death was known -which is not -it would be possible to test this assumption.

Although, the analysis in this paper is very similar to [START_REF] Van Doorslaer | Does inequality in self-assessed health predict inequality in survival by income? Evidence from Swedish Data[END_REF] analysis of the Swedish Survey of Living Conditions, there is one important difference in the interpretation of the results. Van Doorslaer & Gerdtham do not interpret significant main effects of covariates as evidence for reporting bias and thus as a need to worry. For instance, they find significant main effects of income on mortality but no interaction effects between income and self-reported health and conclude: "The fact that the SAH effect does not seem to differ by SES indicators like income or education suggests that there is no systematic adjustment of SAH reporting by SES." (p.1628). The absence of an interaction effect means that being in good rather than fair health has always the same effect on mortality, no matter if you are rich or poor. In other words, the difference in true health between good and fair is the same for all income groups. But it does not mean that good and fair reflect the same levels of health for all income groups. The absence of interaction effects is not sufficient to conclude that comparability of self-assessed health across groups is unproblematic.

The paper is organised as follows. In the next section, I will give a brief description of the data and sample characteristics. Section 3 contains the empirical results. I first show how self-assessed health and mortality relate to the socio-economic characteristics of the respondents. Then I examine the relationship between these characteristics and mortality, conditional on self-assessed health. Strong conditional effects indicate substantial differences in response behaviour across socio-economic groups. Section 4 concludes with some recommendations for future research.

Data

The data are drawn from the years 1992 to 2003 of the SOEP, an ongoing panel survey of the German population. A detailed description is given by the SOEP Group (2001).

Although the SOEP started in 1984, 1992 is the first year in which respondents were asked to self-assess general health on the 5-point "very good" to "very poor" scale. I restrict the sample to German respondents aged 50 and older. I also exclude the guest worker sub-sample because many elderly returned to their home countries and thus dropped out of the SOEP. The sample consists of 4,048 individuals, of which 2,258 (55.3 percent) are women (see Table 1).

Date of death information up to 2001 is available either from the household interviews or from a special status check for drop-outs, in which the status of 90.2% of a grand total of 7,902 drop-outs could be verified (720 respondents died, and year of death is known for 718 respondents). The remaining 9.8 percent have either moved to an unknown new address or died. Unfortunately, the released data does not identify these observations, i.e. it is not possible to distinguish who dropped out of the SOEP and was still alive and who dropped out of the SOEP and whose status could not be verified. .9 years, with women being on average about two years older than men. Socioeconomic status is measured by three variables: equivalised annual net household income (in quartiles), home ownership (as a crude measure of wealth), and schooling (less than high school, high school, and more than high school). The men in the sample have on average higher income and they live more often in owner-occupied housing than the women. One third of the sample are East Germans (they are over-sampled; note that all results presented in this paper are unweighted). Finally, I have generated a dummy variable for early retirement, which is 1 for all men who are younger than 63 and retired and all women who are younger than 60 and retired, and 0 for all others. <about here Table 1> Women are in worse self-assessed health than men. 34.6 percent of the men say that they are in good or very good health, whereas only 27.9 percent of the women do so. As in many other studies, women in the SOEP report worse health but they live longer. As mentioned before, there are only few variables in the SOEP that could be used as contemporaneous quasi-objective health measures, such as doctor visits, hospital stays, and disability status. The relationship between true health and reported health, conditional on these measures, is likely to be affected by our covariates. For example, women tend to go to doctors more often than men independent of their true health status. Richer people might be able to afford more doctor visits, better educated individuals might be more knowledgeable about health issues and thus visit doctors less often in case of minor illnesses, etc. Doctor visits are very much choice dependent, in contrast to hospital stays. The SOEP contains (retrospective) data on the number of hospital stays and the number of nights spent in hospital. However, these data are available only for 1991, not 1992. Still, I will use the number of hospital stays in 1991 F o r P e e r R e v i e w 9 as a proxy for objective health status. The proportion of respondents who have stayed in hospital at least for one day is about 14 percent. Roughly one half of these have stayed up to 14 nights. Sex differences in hospital stays are negligible.

Results

I begin by describing the relationship between the covariates and self-reported health one the one hand and mortality on the other hand. Columns 1 and 2 of 2> The main argument of this paper is that the results of this naive regression cannot be taken at face value. Columns 3 and 4 of Table 2 show odds ratios of logistic regression of being dead by 2003 on characteristics in 1992. Since mortality is an objective indicator of health, the effects of the covariates on mortality should not be much different from their effects on self-assessed health as a subjective measure. At least, they should have the same direction. Indeed, most results are consistent and in line with earlier studies, e.g. the positive correlation between income and wealth on the one side and self-assessed health as well as mortality on the other side. Still, there are some differences between both regressions that are worth being mentioned. First, mortality increases more steeply in age than self-reported health. This is illustrated in Figure 1, where I show predicted proportions of being in poor or very poor health and 11-year mortality by age in 1992. One interpretation of the differences slopes is that individuals get used to decreasing health when they age, so that their reference level for poor health changes.

<about here Figure 1> F o r P e e r R e v i e w 11 Further, it is unclear why East Germans report to be healthier than West Germans, although their life expectancy is about the same in the sample (in fact, it is lower in the population). The mortality of early retirees is only insignificantly larger than the mortality of others, which stands in contrast to their worse self-reports of health. One explanation for these results are differences in reporting behaviour between East and West Germans, or that self-reports of health are endogenous to labour market states.

The next step of the analysis is to estimate the life expectancy of different socioeconomic groups conditional on their self-assessed health. Tables 3a and3b show the estimates of a discrete-time proportional hazard model with a non-parametrically specified baseline hazard [START_REF] Jenkins | Easy Estimation Methods for Discrete-Time Duration Models[END_REF][START_REF] Prentice | Regression analysis of grouped survival data with application to breast cancer data[END_REF], separately for men and women. As mentioned above, respondents who leave the SOEP and whose status in 2003 is unknown are included in the analysis but coded as right-censored at the time of drop-out. Although possible with the available data, I do not let time-varying covariates (in particular: self-assessed health) vary with time. The reason is that I want to predict future mortality on the basis of the information available in 1992, i.e. on the basis of a single cross-section of data. <about here Tables 3a and3b> Tables 3a and 3b contain the results of four different specifications, with and without self-rated health and with and without hospital stays. The reference category for selfassessed health is "fair", which is the middle category. Expectedly, self-assessed health has a strong effect on mortality (see column 2). For example, the hazard rate of a male respondent in very good health is only half as large as the hazard rate of a respondent in the reference category. The hazard rate of a male respondent in very poor health is 2.5 12 times as large as the hazard rate of someone in "fair" health. For women, the effects of poor and very poor health (compared to fair health) are quantitatively similar to those for men. However, the differences between those in good or very good health to those in fair health are smaller than for men and statistically insignificant.

<about here Figure 2> The variation of life expectancy by self-reported health and the sex differences in this variation are illustrated in Figure 2. It shows predicted survival rates for men and women with different self-reported health states and a fixed set of characteristics (West German, married, aged 70, 2 nd income quartile, renter, high school education). For example, the 11-year survival rate of men in very good health was 73.2%. It was 52.0% for men in fair health, and men in very poor health had a survival probability of only 20.5%. Women in very good health had a survival probability of 81.1%, compared to 74.6% if in fair health and 48.5% if in very poor health. Conditional on self-reported health, women live longer than men, but the difference is not uniform across the selfreported health distribution. There is a small difference between men and women in very good self-reported health and a large difference between men and women in very poor health. The distinction between very good and fair is more informative among men than among women. Moreover, it is statistically significant only among men. <about here Figure 3> Figure 3 shows variations in life expectancy by the number of nights spent in hospital in 1991, keeping all other characteristics fixed. Again, conditional on the number of nights spent in hospital, women are less likely to die than men. Note that there is no difference 13 between women who spent up to 14 nights in hospital and those who have not been hospitalised. One explanation for this finding would be that women's short episodes of hospitalisation are frequently due to the treatment of non-lethal chronic diseases.

We now turn to the effects of the covariates. Unsurprisingly, age in 1992 has a positive effect on mortality, but it is interesting to note the difference between the sexes. Male mortality rates increases much faster in age between 50 and 60 than female mortality rates. Between age 60 and 75, the increase is about the same across the sexes and above the age of 75, the increase in male mortality rates is slower then the increase in female mortality rates. These results suggest that much of the difference in mortality between men and women is due to health events in the life of 50 to 60 year olds.

Being married appears to be beneficial especially for men. In fact, the affect is quite large. One reason could be unobserved differences in risk factors (drinking, smoking, and diet) or other health behaviours (such as the likelihood of visiting a doctor conditional on being sick) associated with marital status. Another explanation could be a selection effect if healthier men are more likely to be married (see e.g. [START_REF] Lillard | Marital Status and Mortality: The Role of Health[END_REF]. However, as we have seen above, married men were about as healthy at baseline (in 1992) as unmarried, divorced or widowed men. There is also a clear negative relationship between economic status (measured by equivalised household income in 1992 and home ownership) and mortality. The relationship is a bit weaker for women than for men. Education has a positive effect on survival at all ages. In particular those with less than high school education die earlier than others, but the effect is only weakly statistically significant. Generally, a comparison of the models with and without self-reported health reveals that the coefficients of most covariates barely change when subjective health status or hospitalisation are included in the regression. The inclusion of subjective health usually attenuates the effect of other covariates, but including hospitalisation often has no influence at all on covariates' coefficients. Thus, even if we condition on self-reported health or hospitalisation, age, marital status and income continue to have explanatory power for mortality. This is especially true for men, although this is unlikely a genuine sex difference. If the sample is split further by age, it can be shown that economic status has a larger effect on mortality at young ages than at old ages (see e.g. [START_REF] Deaton | Aging and Inequality in Health and Income[END_REF]. Since women live longer, female mortality rates on average tend to be less affected by economic status than male mortality rates.

The finding that some covariates' effects to not vanish if subjective health is controlled for raises important questions. First, it could mean that self-assessed health is not comparable across different socio-economic groups. For example, the fact that, conditional on self-reported health, mortality increases quite sharply in age, suggests that older respondents tend to overstate their health relative to younger respondents. This happens because of habituation to poor health as one becomes older, i.e. because of a shift in the reference point. An alternative explanation would be cohort effects. It is possible that older cohorts are simply "complaining less" than younger cohorts.

However, given the nature of our data, it is not possible to distinguish age and cohort effects.

Another example is economic status. The fact that rich respondents live longer (conditional on self-reported health) could be explained by a tendency to understate their health relative to poor respondents. This finding becomes relevant when social inequalities in health are analysed on the basis of self-reports. It is likely that inequalities will be understated. Further, the parameter of early retirement is interesting from an economic viewpoint. Several studies have examined whether self reports of health are endogenous to labour market states [START_REF] Bound | Self-Reported versus Objective Measures of Health in Retirement Models[END_REF][START_REF] Dwyer | Health problems as determinants of retirement: are self-rated measures endogenous?[END_REF].

If respondents report to be in poor health partly to justify that they are in early retirement (justification bias), early retirees should survive longer, conditional on selfreported health. The results shown in Tables 3a and3b do not lend support to this assumption. It even seems as if early retirees slightly overstate their health (relative to others). However, the effect is not significant. Qualitatively similar results are obtained when the sample is restricted to respondents who in 1992 were younger than 65.

So far, we have only looked at differences in levels (or health index shifts, in the language of Lindeboom and Van Doorslaer ( 2004)). A further question is whether the relationship between self-assessed health and mortality differs across groups. This can be assessed by estimating the above equations with interaction effects between health and other explanatory variables and test whether the interaction effects are jointly significant. For example, using Swedish data, [START_REF] Van Doorslaer | Does inequality in self-assessed health predict inequality in survival by income? Evidence from Swedish Data[END_REF] find larger gradients for women than for men, and for older than for younger respondents, but no interaction effects between income or education and self-assessed health.

For the sake of brevity, I omit detailed regression results and only show joint tests of interaction terms between self-assessed health and each of the covariates listed in Table 4. The null hypothesis of all interaction terms being equal to zero is rejected for three variables, each time only among women: age, income, and early retirement status. Note that, for women, it was not possible to compute an interaction effect of early retirement <about here Table 4> Significant interaction effects are illustrated in Figures 4 to 6. For those variables where I found significant interaction effects among women, I also show the (generally insignificant) interaction effects for men. The figures show predicted survival rates relative to respondents in fair health (who are thus represented by the horizontal line at 1). Differences across sub-groups, e.g. young and old respondents, in the effect of specific subjective health states on survival rates are thus differences in relative risks.

Figures 4a and4b show the survival rates for 55 year old ("young") and 75 year old ("old") men and women, respectively. Among young respondents, the differences in survival rates between those in very good and those in fair health are small and statistically insignificant. Only men who report poor or very poor health are less likely to survive at all ages. Among older men, there is also a difference in survival rates between those in very good and those in fair health. Although that difference is large in size it is statistically insignificant simply because the number of old men in very good health is small. As their male counterparts, 55-year old women in very poor health are less likely to survive than those in fair or better health. However, the survival probability of younger women in poor health is only slightly smaller than the survival probability of those in fair or better health. This changes when women become older. At age 75, the largest survival gap is between women in poor and very poor health on the one side and women in fair or better health on the other side. The interaction affect Overall, relative survival differences between individuals reporting different levels of self-reported health increase in age. True health differences between young respondents reporting different self-assessed health levels are smaller than true health differences between older respondents reporting different self-assessed health levels. In fact, as far as mortality is concerned, the distinction between very good, good, and fair is more or less meaningless among the young.

<about here Figures 4 to 6>

Figures 5a and5b show interaction effects of self-reported health with income by comparing predicted survival rates of 70 year old men and women with income below and above the median ("poor" and "rich"), respectively. Statistically, the mortality gap between rich and poor men in different health states is insignificant. The only visible difference between rich and poor men in relative survival is among the few respondents in very good health. In contrast to men, interaction effects for women are statistically significant. First, being in very good health has a positive effect on survival among the poor but no effect among the rich. Second, poor and very poor health have about the same effect on mortality among poorer women. Among richer women, the relative effect of poor health is smaller.

Finally, Figures 6a and6b show the interaction effects of early retirement and selfassessed health for 60 year old men and women, respectively. As explained above, the model estimated for women suffers from complete separation. Hence I omit survival F o r P e e r R e v i e w 18 rates for respondents in very good health. Again, we cannot see much difference in relative survival rates among early retired men and their non-retired counterparts. However, early retired women in very poor health are much less likely to survive than non-retirees -relative to the counterparts in fair health.

Discussion

This paper examines the structure of the relationship between self-assessed general health and subsequent mortality among respondents aged 50 and over using data from the German Socio-Economic Panel. In general, self-assessed health proves to be a strong predictor of mortality. However, conditional on self-assessed health, other covariates such as sex, age, or income still have strong independent effects on mortality, which are often similar in size to their unconditional effects. This should not be case if self-assessed health was a valid and unbiased proxy measure for true health. Rather, self-assessed health should absorb many of the health risks associated with these covariates.

The use of self-assessed health as a proxy for true health can thus give rise to misleading inferences about social inequalities in health or the effects of health on retirement, health care use, etc. Whether that is the case depends on the reason for the independent effects of age, sex, or income on mortality. There are two complementary explanations. First, lethal events that are unexpected and uncorrelated with current health might vary systematically by these covariates. If that is the case, no harm is done, because we can still assume that self-reports reflect true health. For example, if men are more likely to be killed in a car accident, they will have a higher mortality, independent of their true health status. The question is whether such events are frequent enough to 19 explain the mortality gap between equally healthy men and women. Information on chronic conditions or cause of death would be of great help to answer this question [START_REF] Case | Sex differences in morbidity and mortality[END_REF]. Another example is the effect of income or wealth. A wealthy person who is in the same true health status as a poor person might be able to "buy time", for example because she can afford better treatment and thus increase her lifeexpectancy. However, the public health systems in many European countries, including Germany, are so generous that, conditional on true health, wealth should not matter much for life expectancy.

The second -more damaging -explanation is that response behaviour differs across different groups, i.e. individuals with the same true health status have different reference points against which they judge their health. Respondents do not perceive the self-assessed health scale as absolute. They may be likely to report "very poor" health only if they feel they are much less healthy than others of the same sex, age, education, or income. In fact, some surveys (e.g., the BHPS) take that into account when they ask respondents to rate their health relative to other people of their own age. The fact that self-rated health increases in age in the BHPS, conditional on a number of quasiobjective measures, is strong support for the shifting reference point explanation [START_REF] Groot | Adaption and scale of reference bias in self-assessments of quality of life[END_REF]. A related argument that applies to cross-cultural comparisons of health refers to differences in habitual language use. For instance, "excellent" is a term that is used in everyday parlance in the Anglo-Saxon world, but Germans would often consider this attribute as an ironic exaggeration [START_REF] Sadana | Comparative analyses of more than 50 household surveys on health status[END_REF][START_REF] Jürges | Cross-country differences in self-reported health. Ch. 3.4[END_REF]. Objective health measures and maybe anchoring vignettes are needed to ascertain the relevance of response styles. A more promising option is to make self-assessed health comparable across groups on the basis of specialised health surveys that provide subjective and detailed objective measures of health (and possibly vignettes). This appears to be self-evident in crossnational research, but it surely will also help to improve analyses using national surveys.

  has recently been introduced into Grossman-type models of health investments by

  Only verified deaths are coded. Of the 4,084 observation in my sub-sample, 857 are drop-outs not coded as dead. These observations are treated as right-censored at the time they leave the panel. 25 percent of the men and 21 percent of the women were dead in 2003. Another 21 percent left the sample for other reasons. The life expectancy of those who died between 1992 and 2003 was on average 5.5 years. The average age (in 1992) of my sample is

  self-reported health. The model suffered from separation because all early retired women in very good health survived.

  age is basically this change in the (substantive) significance of poor self-reported health for survival.

  implications of the findings of the present and other studies for empirical research on the causes and consequences of health? It is quite likely that self-assessed health is comparable only within narrowly defined socio-economic groups. This suggest to use subjective health measures only in within-subsample analyses. Apart from likely problems with small sample sizes, it is yet unclear which characteristics should be used to split the sample. Sex and age are the most likely candidates, but beyond that there appears to be no generally accepted set of variables. Moreover, the restriction to different subsamples precludes to answer interesting research questions across subgroups.

  Table 2 contain estimated odds ratios of a "naive" logit regression of being in poor or very poor selfassessed health on age, marital status, income, home ownership, education, region, early retirement, and hospital stays, separately for men and women. Drop-outs, i.e. respondents whose mortality status in 2003 is unknown have been dropped from these regressions. Odds ratios larger than one indicate decreasing self-assessed health. Note that the respondents' age enters the empirical model as a piecewise linear function. Age1 measures the age effect in the 50 to 60 interval, Age2 measures the age effect in the 60 to 75 interval, and Age3 measures the age effect above age 75.

	10
	• education has a strong positive and significant relation to self-reported health among
	men, but virtually no effect among women
	• early retirees are muss less healthy than others
	• East Germans are much healthier than West Germans
	• short episodes of hospitalisation have only little effect on self-assessed health,
	whereas longer episodes have very large effects
	<about here Table
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	R e
	v i
	One purpose of this regression is to describe the differences in self-reported health status in the sample. According to the estimated model e w
	• older respondents are less healthy than younger respondents
	• married and unmarried respondents are equally healthy
	• the poorest income quartile is the least healthy. Health differences across the
	remaining quartiles are comparatively small
	• wealth (i.e. home ownership) has a weak and insignificant positive relationship with
	self-reported health.

Page 13 of 33

  Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

	Submitted Manuscript
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	e w

Table 1 :

 1 Sample Description

	Variable

Table 2 :

 2 Logistic regressions of being in poor or very poor health and 11-year mortality, by sex

	Poor self-assessed health	Mortality
	Men	Women	Men	Women

Table 3a : Discrete-time proportional hazards model of men's mortality, exponential form

 3a

				(1)	(2)	(3)	(4)
	Age1 [min(Age, 60)]			1.126**	1.127**	1.123**	1.125**
				(0.030)	(0.030)	(0.031)	(0.031)
	Age2 [max(min(Age, 75), 60) -60]	1.095**	1.082**	1.090**	1.080**
				(0.014)	(0.014)	(0.014)	(0.014)
	Age3 [max(Age, 75) -75]	1.064**	1.061**	1.071**	1.064**
				(0.017)	(0.017)	(0.018)	(0.018)
	Married			0.730*	0.679**	0.721**	0.682**
				(0.090)	(0.082)	(0.090)	(0.083)
	Income: 1 st quartile			1.000	1.000	1.000	1.000
	Income: 2 nd quartile			0.812	0.857	0.837	0.869
				(0.108)	(0.115)	(0.111)	(0.116)
	Income: 3 rd quartile			0.672**	0.716*	0.682*	0.707*
				(0.101)	(0.108)	(0.102)	(0.106)
	Income: 4 th quartile Home owner F o	0.507** (0.089) 0.830+ (0.082)	0.573** (0.102) 0.865 (0.086)	0.530** (0.092) 0.828+ (0.082)	0.584** (0.103) 0.859 (0.085)
	Less than high school	r	1.308+	1.207	1.450*	1.316+
	High School More than high school Early Retiree	(0.191) 1.000 0.847 (0.125) 1.108 e P e (0.189) r	(0.177) 1.000 0.872 (0.129) 1.044 (0.177)	(0.214) 1.000 0.864 (0.127) 1.096 (0.188)	(0.196) 1.000 0.873 (0.129) 1.048 (0.179)
	East Germany Health: very good Health: good			0.799+ (0.102) R e v 0.810+ (0.104) 0.503+ (0.195) 0.726* (0.100) i	0.882 (0.114)	0.873 (0.113) 0.523+ (0.203) 0.769+ (0.107)
	Health: fair Health: poor Health: very poor				1.000 1.481** (0.185) 2.518** e w		1.000 1.358* (0.173) 2.262**
					(0.343)		(0.316)
	Nights in hospital: 0					1.000	1.000
	Nights in hospital: 1 to 14				1.566**	1.301
						(0.272)	(0.234)
	Nights in hospital: 15 to 35			1.834**	1.575**
						(0.310)	(0.269)
	Nights in hospital: more than 35			3.163**	2.226**
						(0.581)	(0.426)
	Year dummies			yes	yes	yes	yes
	Observations			14,882	14,882	14,882	14,882
	Note -Standard errors in parentheses +p < 0.10; * p < 0.05; ** p < 0.01

Table 3b : Discrete-time proportional hazards model of women's mortality, exponential form

 3b

						27
			(1)	(2)	(3)	(4)
	Age1 [min(Age, 60)]		1.057+	1.052	1.057+	1.052+
			(0.032)	(0.032)	(0.032)	(0.032)
	Age2 [max(min(Age, 75), 60) -60]	1.103**	1.099**	1.099**	1.096**
			(0.015)	(0.015)	(0.015)	(0.015)
	Age3 [max(Age, 75) -75]	1.145**	1.128**	1.148**	1.133**
			(0.016)	(0.016)	(0.016)	(0.016)
	Married		0.959	0.916	0.972	0.923
			(0.110)	(0.105)	(0.112)	(0.106)
	Income: 2 nd quartile		0.855	0.977	0.825	0.923
			(0.104)	(0.122)	(0.100)	(0.115)
	Income: 3 rd quartile		0.626**	0.717*	0.596**	0.673**
			(0.094)	(0.109)	(0.089)	(0.103)
	Income: 4 th quartile		0.563**	0.661*	0.573**	0.659*
			(0.107)	(0.128)	(0.109)	(0.127)
	Home owner Less than high school F o High school r	0.952 (0.097) 1.196+ (0.122) 1.000	0.975 (0.100) 1.158 (0.119) 1.000	0.954 (0.097) 1.174 (0.121) 1.000	0.969 (0.100) 1.142 (0.118) 1.000
	More than high school Early Retiree	1.059 (0.254) 1.343 (0.364) e P e	1.032 (0.248) 1.210 (0.328)	0.987 (0.239) 1.378 (0.374)	0.965 (0.234) 1.248 (0.339)
	East Germany		0.949 r	1.025	0.934	0.999
			(0.112)	(0.122)	(0.110)	(0.119)
	Health: very good Health: good Health: fair		0.759 (0.277) 0.812 (0.127) 1.000 v R e i		0.779 (0.284) 0.822 (0.128) 1.000
	Health: poor Health: very poor			1.412** (0.167) 2.479** (0.351) e w		1.339* (0.160) 2.201** (0.320)
	Nights in hospital: 1 to 14				0.898	0.816
					(0.172)	(0.158)
	Nights in hospital: 15 to 35			2.070**	1.703**
					(0.359)	(0.303)
	Nights in hospital: more than 35			2.821**	2.205**
					(0.557)	(0.446)
	Year dummies		yes**	yes**	yes**	yes**
	Observations		18,764	18,764	18,764	18,764
	Note -Standard errors in parentheses +p < 0.10; * p < 0.05; ** p < 0.01

Table 4 : Interaction effects of self-reported health with covariates, Chi-squared test statistics

 4 Predicted proportion of respondents in poor or very poor health and predicted 11-year mortality (by sex). Predicted survival rates of 70 year olds, by sex and self-reported health (Source: SOEP 1992-2003) Predicted survival rates of 70 year olds, by sex and hospitalisation (Source: SOEP 1992-2003) Interaction effects of self-assessed health and age on predicted survival rates of men (Source: SOEP 1992-2003) Interaction effects of self-assessed health and age on predicted survival rates of women (Source: SOEP 1992-2003) Interaction effects of self-assessed health and income on predicted survival rates of men (Source: SOEP 1992-2003) Interaction effects of self-assessed health and income on predicted survival rates of women (Source: SOEP 1992-2003) Interaction effects of self-assessed health and early retirement on predicted survival rates of men (Source: SOEP 1992-2003) Interaction effects of self-assessed health and income on predicted survival rates of women (Source: SOEP 1992-2003)

			Men			Women
		Chi-	DF	p-value	Chi-	DF	p-value
		Squared			Squared
	Sex	0.58	4	0.965
	Age	3.48	4	0.481	10.2	4	0.037
	Married	2.66	4	0.616	2.88	4	0.579
	Income (below vs. above median)	5.16	4	0.271	9.51	4	0.050
	Home ownership	3.51	4	0.477	1.34	4	0.855
	Education (high school and above)	2.72	4	0.606	1.06	4	0.900
	Early retirement	0.61	4	0.962	7.73	3 a)	0.052
	East Germany	0.58	4	0.965	1.28	4	0.865
	Note -a) One category predicted failure prefectly

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UKSubmitted Manuscript


Florian Hoeppner

Thorsten Neumann

Florian Höppner

Christian Melzer
email: melzer@iiw.uni-bonn.de.

Manfred J M Neumann

Athanasios Orphanides

Kenneth L Judd

J Huston Mcculloch

Changing Effects of Monetary Policy in the U.S. Â Evidence from a Time-Varying Coefficient VAR

Keywords: monetary transmission, vector autoregression, non-linear time series models, time-varying coefficients JEL classification: E52, E32, C52 vector autoregression, monetary transmission, non-linear time series models, time-varying coefficients

Changing Effects of Monetary

Policy in the U.S. -Evidence from a Time-Varying Coefficient VAR

Introduction

Has the impact of U.S. monetary policy on output changed systematically over time? Moreover, are the effects of monetary policy symmetric or asymmetric over the business cycle?

These issues have received increasing interest in the recent macroeconometric literature. By applying a time-varying coefficient vector autoregressive model (TVC-VAR) in the present paper we are able to tackle both these issues within one econometric framework. In comparison to other non-linear empirical models that have been used to study time-varying effects of monetary policy (e.g., the Markov-Switching model or the smooth-transition VAR model), the TVC-VAR constitutes an interesting alternative. It is comparatively flexible and it imposes as few as possible restrictions on the data. The empirical model is thereby well suited for a -to a certain extent -"unprejudiced" look at the data, which lies in the VAR-tradition of analysing the effects of policy shocks. Some papers such as [START_REF] Cogley | Evolving Post World War II Inflation Dynamics[END_REF] and Cicarelli and Rebucci (2002) recently adopted Bayesian frameworks for TVC-VAR analysis of monetary policy that are related to our approach. Our results suggest a decreasing impact of U.S. monetary policy over time as well as considerable asymmetries over the business cycle, with stronger effects of monetary policy during a recession.

At least two theories of the monetary transmission suggest changing effects of monetary policy over time. The 'cost channel' presented by, e.g., [START_REF] Barth | The Cost Channel of Monetary Transmission[END_REF] claims that monetary policy effects are transmitted through the supply side of the economy by affecting the working capital of enterprises. They develop three factors that may account for a weakening of the real effects of monetary policy through this channel: (i) Financial innovations and deregulation in the U.S. financial system increased the availability of working capital. (ii) The change to a floating exchange rate system after the breakdown of the Bretton Woods system. This counterbalances the directly increasing costs of working capital after a monetary tightening by a reduction in costs of imported materials. (iii) The Federal Reserve monetary policy In addition to the cost channel, systematic changes of monetary policy effects over time can be derived from the 'credit channel', which is typically divided into the 'bank lending channel' and the 'balance sheet channel'.1 Both channels point to changes in the private sector's financial structure as potential sources of changing effects of monetary policy. In particular, as financial innovations and the integration of financial markets make it easier to raise funds on capital markets, the dependence of the private sector on bank credit decreases and the bank lending channel likely becomes less effective.

Besides theories of monetary transmission, changes in the way monetary policy is conducted can alter the real effects of monetary policy shocks. Boivin and Gianonni (2002) employ a stylized model to argue that a decline in the real effects of monetary policy can be explained by a central bank that has an increasing preference for output stabilization in the face of supply and demand shocks and therefore makes more efficiently and more rigorously use of its own instruments.2

The second issue to be addressed in the empirical analysis are potential asymmetries of monetary policy over the business cycle. On the one hand, business cycle dependent effects of monetary policy can be motivated from a convex aggregate supply curve. In the flat part of the supply curve where output is relatively low as for example in a recession, a shift in the demand curve has a larger impact on output and a smaller on prices in contrast to the steeper part of the supply curve, where output is relatively high. This enhances the problems created by informational asymmetries. In a boom, when firms and households have a relatively high net worth, policy actions are less effective than in a recession when the net worth is relatively low.

As has been noted above, so far empirical studies have focussed on one of the two issues alone. [START_REF] Barth | The Cost Channel of Monetary Transmission[END_REF] and Boivin andGiannoni (2002, 2003), for example, estimate

VARs for different sub-samples and find the average effect of monetary policy on output to decrease in the U.S. Moreover, concerning potential business cycle asymmetries, recent empirical evidence for the U.S. by [START_REF] Weise | The Asymmetric Effects of Monetary Policy: A Nonlinear Vector Autoregression Approach[END_REF], [START_REF] Garcia | Are the Effects of Monetary Policy Asymmetric?[END_REF] and [START_REF] Lo | Is the Response of Output to Monetary Policy Asymmetric? Evidence from a Regime-Switching Coefficients Model[END_REF], among others, suggests stronger effects of monetary policy in a recession than in a boom.

In the present paper, we take up both issues raised and analyse them in a unifying empirical framework. We use a standard VAR framework and following [START_REF] Neumann | Expansionary Fiscal Consolidations: Evidence from Time-Varying Coefficient VARs for Germany[END_REF], we allow for time-varying coefficients following a random walk. This introduces an empirical framework referred to as TVC-VAR which is able to capture a potentially changing structure of the economy over time. In particular, time-varying impulse responses are generated that visualise the nature of this structural change over time. From time-varying impulse response estimates new insights can be gained concerning the evolution of monetary policy effects over time as well as potential asymmetries over the business cycle. Section 2 introduces the empirical framework. Section 3 presents our empirical findings and Section 4 concludes. 4 See [START_REF] Hubbard | Is There a Credit Channel for Monetary Policy?" Federal Reserve Bank of St[END_REF] and [START_REF] Bernanke | The Financial Accelerator and the Flight to Quality[END_REF]. 2 Empirical Methodology -The TVC-VAR Framework

In recent years, a huge amount of empirical models have been developed to account for structural breaks and potential non-linearities including regime-switching models, threshold autoregressive models as well as state-space models with time-varying coefficients.5 Essential for the choice of the appropriate modelling framework thereby is the type of coefficient variation that is most likely for the phenomenon under investigation. Empirical findings based on simulated data from [START_REF] Neumann | Time-varying coefficient models: A comparison of alternative estimation strategies[END_REF] suggest that time-varying coefficient models with random walk coefficients dominate alternative approaches to time-varying estimation. Moreover, a model with random-walk coefficients may be appropriate even in the presence of timeinvariant coefficients because model estimates turn out to be comparatively stable in this case.

In this paper we follow [START_REF] Jiang | A Time-Varying Coefficients Vector AR Modeling of Nonstationary Covariance Time Series[END_REF] and [START_REF] Neumann | Expansionary Fiscal Consolidations: Evidence from Time-Varying Coefficient VARs for Germany[END_REF] and extend univariate time-varying estimation to VAR analysis. Time-varying impulse responses derived from our model estimates allow us to investigate the real effects of monetary policy shocks over time. In the following, a brief sketch of the methodology is presented, see [START_REF] Neumann | Expansionary Fiscal Consolidations: Evidence from Time-Varying Coefficient VARs for Germany[END_REF] for a more detailed exposition.

The Model Set-Up Consider the following reduced form of a VAR with p lags and n en-

dogenous variables: 0, 1, 1 , ... t t t t pt t p t Y A A Y A Y U - - = + + + + .
(1)

In this set-up, for every t of the sample coefficient matrices A 0,t , A 1,t , … ,A p,t and a variancecovariance matrix Σ U,t are estimated, where U t is distributed as U t ~ N(0, Σ U,t ). Collecting the coefficient matrices in matrix A t = (A 0,t , A 1,t , … , A p,t ), and defining t B = vec(A t ) and X t = (1

⊗ I k , Y' t-1 ⊗ I k , … , Y' t-p ⊗ I k )
, the model can be written as In order to get reasonable estimates of the coefficients from the limited amount of data points available, stochastic constraints are imposed. More specifically, the time variation of the coefficients is specified by assuming that the elements of t B follow independent random walks,

t t t t Y B X U = + . (2)
1 t t t B B W - = + (3)
with W t ~ N (0, Σ W ) and Σ W being diagonal. This restriction constitutes the Gaussian 'smoothness prior' distribution on the time history of the VAR coefficients. 6

In order to enable estimation of the n-dimensional system with time-varying coefficients, following [START_REF] Jiang | A Time-Varying Coefficients Vector AR Modeling of Nonstationary Covariance Time Series[END_REF] and [START_REF] Neumann | Expansionary Fiscal Consolidations: Evidence from Time-Varying Coefficient VARs for Germany[END_REF], a Cholesky recursive structure is imposed on the system, allowing to estimate the VAR equation by equation. Assuming that the structural form of the VAR follows a recursive structure, equation ( 1) can be written as 0,

...

t t t t t p t t p t Y C C Y C Y V - - Γ = + + + + , (4)
where the structural residuals are distributed as V t ~ N (0, Σ V ), and Σ V being diagonal. The lower triangular matrix Γ t captures the recursive contemporaneous interactions of the endogenous variables. As Σ V is a diagonal matrix, the equations of model ( 4) can be estimated equation by equation yielding estimates of the structural coefficient matrices C 0,t , C 1,t ,…, C p,t as well as Γ t , and the variance-covariance matrix Σ V .

6 Alternatively an autoregressive structure like

1 t t t B B W α - = +
could have been imposed, where 0 < α < 1.

Simulations however show that the random walk model as a general specification captures several potential time paths of gradual coefficient changes quite well. Note, however, that the specification by construction imposes a smooth path for the coefficients. As a consequence, the model behaves badly when the underlying coefficient process exhibits discrete single shifts. An appropriate model to capture discrete stochastic shifts may be the Markov-Switching model, see, e.g., [START_REF] Garcia | Are the Effects of Monetary Policy Asymmetric?[END_REF]. The matrices A i,t and Σ U,t as well as the residuals U t of the reduced form can be recovered using the following equations:

1 , , 1 1 1 , ( )', . t i t t i t U t V t t t t A C U V - - - - = Γ Σ = Γ Σ Γ = Γ i = 1, …, p
Estimating and Analyzing the Model For estimating (4) equation by equation each equation of the model is written as

' t t t t y = x β + v , ( 5
)
where t y is the dependent variable and t x and t β are vectors collecting the variables and the coefficients of a single equation of model ( 4). ( 5) constitutes the measurement equation of the state-space representation of the model, where the respective transition equation is given by 3 Empirical Results

1 t t t w β β - = + , (6)
We apply the TVC-VAR methodology to a standard three variables VAR model of the U.S.economy, consisting of GDP (deflated with the consumer price index and in logs), consumer prices (in logs) and the federal funds rate. All series are at the quarterly frequency, running from 1962:1 to 2002:2. 7 The federal funds rate is used as a measure of monetary policy for the whole sample range. This approach is widely used in the empirical literature, for a discussion see, e.g., Bernanke and Mihov (1998). 8 Unit root tests indicate that output and prices are I(1), whereas the federal funds rate is I(0), see Table 1 for details. Therefore, except for the federal funds rate the model is estimated in first differences. With this specification we follow, among others, [START_REF] Rudebusch | Policy Rules for Inflation Targeting[END_REF]. No statistical criterion is available yet for the choice of the lag order in the TVC-VAR case. The results, however, are comparatively robust to alternative choices. Hence, we restrict the presentation of estimation results to a lag order of four. Finally, as the estimation and identification of the model relies on the recursive Cholesky structure, we follow the standard procedure to order the monetary policy instrument last in the VAR, after GDP and prices. We choose this ordering because it has the strongest theoretical background given that GDP and prices are unlikely to react simultaneously to interest rate shocks. Nonetheless, we considered alternative orderings in order to check the robustness of reported results. In fact, we found the results to be rather robust to alternative orderings of the monetary policy instrument. This finding indicates that a low degree of simultaneous correlations among the examined variables.

7 All data are from the IMF's "International Financial Statistics": Gross Domestic Product, series code 11199b.c. Consumer Price Index, series code 11164. Federal Funds rate, series code 11160b. 8 Even though the Federal Reserve changed it's operating procedure in the period of 1979-1982, the funds rate was still closely connected with the Federal Reserve monetary strategy, on this see, e.g., [START_REF] Cook | Determinants of the Federal Funds Rate[END_REF], [START_REF] Goodfriend | Interest Rates and the Conduct of Monetary Policy[END_REF] and [START_REF] Bernanke | Central Bank Behaviour and the Strategy of Monetary Policy: Observations from Six Industrialized Countries[END_REF]. Among others [START_REF] Sims | Interpreting the Macroeconomic Time Series Facts. The Effects of Monetary Policy[END_REF] and [START_REF] Clarida | Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory[END_REF] have used the funds rate for comparable sample ranges in a VAR-analysis as monetary policy instrument. From TVC-VAR estimates we derive an impulse response function for every point of time over the whole sample period. Figure 1 presents the accumulated impulse response at several horizons of real GDP to a monetary policy shock (one percentage point increase in the federal funds rate) as it evolves over the estimation period. It contains four graphs (a) to (d) that indicate how the accumulated impulse responses at a horizon of 4, 8, 12 and 16 quarters evolve over the sample period, respectively. For reasons of an illustrative comparison every graph also plots as a dashed line the accumulated impulse response at the respective horizon from the linear VAR specification as well as 10% error bands from the latter specification. This is intended to serve as a rough guideline to judge the degree to which the time-varying specification departs from the linear one. Yet, it is important to stress that this does not constitute a formal test for non-linearity, as the estimates from the linear specification are biased in the presence of structural change. 9 Concerning the first issue to be analysed in this paper, namely the stability of monetary policy effects over time, the results suggest that the effects of monetary policy shocks have become weaker over time. Most clearly this 'trend' is visible in the reaction of GDP to the monetary policy shock after four quarters. The impact today is almost half as strong as it was in the 1970s. While at the beginning of the 1970s a one percentage point increase in the federal funds rate led to a decrease in real GDP after four quarters of roughly 0.5 percent, in the second half of the 1990s this effect was down to around 0.2 percent, though increasing again at the end of the sample period. It is also interesting to note that at the four quarter horizon the impulse response moves out of the linear error bands at the beginning of the 1980s. As has been noted above, this may indicate a structural break at the beginning of the 1980s. A linear specification over the whole sample period hence may systematically overestimate the effect of a monetary policy shock on output.

The finding of a declining impact of monetary policy shocks over time supports the cost channel and the credit channel of monetary transmission. Both refer to changes in the finan-9 Computing error bands for the time-varying impulse responses on the other hand is still an unresolved issue. cial structure that translate into a weakening of monetary policy effects. Additionally, our findings support theories that are based on a change in the monetary policy reaction function causing the effects of monetary policy to decline. To differentiate between the various theories is beyond the scope of the TVC-VAR framework. In fact, it requires the use of more structural models that help to disentangle the causes of a declining impact of monetary policy shocks over time. However, the gradual change in the real effects of monetary policy over the last 40 years might tentatively speak in favor for theories of monetary transmission linking the changes over time to gradually changing economic structures, as far as monetary policy seems to be more likely to be subject to abrupt regime shifts rather than to a gradual change.

Empirically our results are consistent with the evidence found in Boivin andGiannoni (2002, 2003) and [START_REF] Barth | The Cost Channel of Monetary Transmission[END_REF]. Boivin andGiannoni (2002, 2003) estimate a linear VAR with different sub-samples and find that the response of output to a monetary policy shock has become weaker since the beginning of the 1980s, similar results at the industry level can be found in [START_REF] Barth | The Cost Channel of Monetary Transmission[END_REF]. Also, the empirical evidence on changes in the Federal Reserve's monetary policy reaction function presented by, e.g., [START_REF] Taylor | A Historical Analysis of Monetary Policy Rules[END_REF], [START_REF] Clarida | Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory[END_REF] is consistent with our results. They find an increased preference for output stabilization in the conduct of monetary policy by the Federal Reserve which might cause monetary policy effects to decline as our estimates suggest. 10

The second issue we are interested in is whether there is a business-cycle dependency in the effects of monetary policy. Therefore, we compare our results with the U.S. recession periods.

In Figure 1 -10 -10 asymmetry became much weaker since the mid 1980s. This observation fits to the available evidence of an increased efficiency in monetary policy in the sense that the Federal Reserve gained more experience in steering the economy through the business cycle.

The second part of our empirical findings are again in line with economic theory, namely with the credit channel and with models of convex supply curves. Moreover, our study confirms the results of other empirical studies that analyse asymmetries of the effects of monetary policy. To mention only three important studies, [START_REF] Garcia | Are the Effects of Monetary Policy Asymmetric?[END_REF] and [START_REF] Lo | Is the Response of Output to Monetary Policy Asymmetric? Evidence from a Regime-Switching Coefficients Model[END_REF] find that monetary policy is stronger during recessions using the Markov-Switching framework introduced by [START_REF] Hamilton | A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle[END_REF], while [START_REF] Weise | The Asymmetric Effects of Monetary Policy: A Nonlinear Vector Autoregression Approach[END_REF] finds the same result applying a non-linear smooth transition VAR framework.

Conclusion

In the present paper we address two questions: (i) Have the real effects of US monetary policy changed over the last 40 years? (ii) Are the real effects of monetary policy asymmetric over the business cycle? To investigate these two issues, we apply a standard three-variable VAR while allowing for time-varying coefficients in this model. From this we derive time-varying impulse responses to investigate the response of output to monetary policy shocks over time.

Two findings emerge from our empirical analysis. First, the impact of monetary policy shocks steadily decreased since the 1960s. This finding supports the cost channel and the credit channel of monetary transmission as well as theories pointing to an increased preference to stabilize output fluctuations by the central bank, leading to a decline in real effects of monetary policy. Second, we find that monetary policy effects are asymmetric over the business cycle, where monetary policy is stronger during recessions. This is in support of the credit 10 The issue of a changing conduct of U.S. monetary policy, i.e. how and if the Federal Reserves's reaction function has changed in the last 40 years is still controversial. See, e.g., [START_REF] Cogley | Evolving Post World War II Inflation Dynamics[END_REF], [START_REF] Orphanides | Monetary Policy Rules and the Great Inflation[END_REF][START_REF] Orphanides | Monetary Policy Rules, Macroeconomic Stability and Inflation: A View from the Trenches[END_REF], [START_REF] Söderlind | What if the Fed Had Been an Inflation Nutter?[END_REF] and Österholm (2005). The interest rate rule states that the central bank wishes to stabilize output around the steady state with φ (and φ > 0) representing the strength of output stabilization. In addition, the pol- icy rule in equation (A.4) includes a disturbance term t ε representing the unsystematic part of monetary policy. In fact, this disturbance term can be thought of as the monetary shock with which impulse-response analysis in a VAR framework of monetary policy is conducted.

F

Solving the model for output t y leads to equation (A.5). The y-axis gives the percentage impact on output of a one unit shock.

  1960s and 1970s were often accompanied with credit control actions leading to a non-price rationing of working capital.

  estimates of the coefficients β 1 ,…,β T can be generated by applying the Kalman filter routine to every single equation given by (5), see Jiang and Kitagawa for an exposition of the Kalman filter for the TVC-VAR application. The model is estimated as outlined in detail in[START_REF] Neumann | Expansionary Fiscal Consolidations: Evidence from Time-Varying Coefficient VARs for Germany[END_REF], using the EM algorithm to find maximum likelihood estimates of the hyperparameters to initialise the Kalman filter. Alternatively Bayesian approaches in the tradition of[START_REF] Doan | Forecasting and conditional projection using realistic prior distributions[END_REF] could be adopted for model estimation. A number of papers such as[START_REF] Cogley | Evolving Post World War II Inflation Dynamics[END_REF] andCicarelli and Rebucci (2002) applied Bayesian techniques in the context of monetary policy analysis recently. We derive timevarying impulse responses using the Generalised Impulse Response approach of[START_REF] Koop | Impulse Response Analysis in Nonlinear Multivariate Models[END_REF]. In contrast toKoop et al. the present analysis assumes that once a shock has occurred there is no feedback of this shock to the coefficients of the model, coefficient variation thus is exogenous.

  the shaded areas correspond to the NBER recession periods. It is clearly evident from Figure 1 that monetary policy tends to have stronger effects during recessions. This result is particularly pronounced at the longer impulse response horizons, most strongly during the recessions 1973:4 until 1975:1 and in the early 1980s. In addition, we observe that this

  Figure 1: Accumulated Impulse Responses of GDP to a Monetary Policy Shock

Page 18 of 19 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

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Table 1 : Unit Root tests ADF Test a KPSS Test b Variable

 1

		Test statistic	Test statistic
		(Specification c )	(Specification c )
	GDP	-3.436* (c,t)	0.159** (c,t)
	∆GDP	-8.585*** (c)	0.216	(c)
	CPI	-1.618	(c, t)	0.290*** (c,t)
	∆CPI	-2.189	(c)	0.326 (c)
	Interest rate	-2.654* (c)	0.264 (c)
		H 0 : Existence of a unit root	H 0 : Stationarity
		* denotes signifcance at 10% -level
		** denotes signifcance at 5% -level

*** denotes signifcance at 1% -level a Augmented Dickey-Fuller Test with lag selection according to the Schwartz criterion; b Kwiatkowski-Phllips-Schmidt-Shin Test; c c =constant, t = trend. Page 19 of 19 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

See, e.g., Hubbard (1995) for an overview over the credit channel.[START_REF] Bernanke | Is it Money or Credit or Both or Neither? Credit, Money and Aggregate Demand[END_REF] present a model of the bank lending channel, whereas[START_REF] Bernanke | The Financial Accelerator and the Flight to Quality[END_REF] model the balance sheet channel. For empirical evidence on the bank lending channel in the U.S. see, e.g.,[START_REF] Gertler | Monetary Policy, Business Cycle Behavior, and the Behavior of Small Manufacturing Firms[END_REF],[START_REF] Kashyap | What do a Million Observations on Banks Say About the Monetary Transmission of Monetary Policy[END_REF] and[START_REF] Nilsen | Trade Credit and the Bank Lending Channel[END_REF].

2 See the Appendix for a presentation of the model byBoivin and Gianonni (2002

).3 [START_REF] Ball | Asymmetric Price Adjustment and Economic Fluctuations[END_REF] use a hybrid framework of time and state-contingent price adjustment rules under the assumption of menu costs to derive a convex supply curve, see also[START_REF] Caballero | Price Rigidities, Asymmetries, and Output Fluctuations[END_REF].

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

See, e.g., Granger and Teräsvirta (1993) for a comprehensive survey.

Appendix

With the aid of a small stylized model this argument can be illustrated. The model proposed by Boivin and Gianonni (2002) consists of two central equations and is a stripped down version of a standard New Keynesian model as used for the analysis of monetary policy by [START_REF] Mccallum | An Optimizing IS-LM Specification for Monetary Policy and Business Cycle Analysis[END_REF] and [START_REF] Clarida | The Science of Monetary Policy: A New Keynesian Perspective[END_REF]

3)

The second central equation of this model represents central bank behavior. This is given in equation (A.4) by a simple Taylor-style interest rate rule. 12 For expository purposes Boivin 11 Usually a third equation is included in these models, providing the inflation adjustment process with inflation adjustment derived from optimizing firms in an environment of monopolistic competition. This equation is omitted to focus on the issue in question.

12 See [START_REF] Taylor | Discretion versus Policy Rules in Practice[END_REF] Page 13 of 19

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript